math refresher* session 1 - weeblyserviceandscience.weebly.com/uploads/3/7/7/4/...thus the net...

$: MATH REFRESHER* session 1 - Weeblyserviceandscience.weebly.com/uploads/3/7/7/4/...Thus the net profit is $56 2 $50 5 $6. The fractional profit is $56 − $50 _____ $50 5 $6 ____ $50$
173

MATH REFRESHER* session 1

*Note: Many of the examples or methods can be done with a calculator, but it is wise for you to know how to solve problems without a calculator.

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174

Fractions, decimals, percentages, deviations,

Ratios and Proportions, Variations, and Comparison of Fractions

Fractions, decimals, percentagesThese problems involve the ability to perform numerical operations quickly and correctly. It is essential that you learn the arithmetical procedures outlined in this section.

101. Four different ways to write “a divided by b” are a 4 b, a __ b , a : b, b ⟌

__ a .

Example: 7 divided by 15 is 7 4 15 5 7 ___ 15 5 7 : 15 5 15 ⟌__ 7 .

102. The numerator of a fraction is the upper number and the denominator is the lower number.

Example: In the fraction 8 ___ 13 , the numerator is 8 and the denominator is 13.

103. Moving a decimal point one place to the right multiplies the value of a number by 10, whereas moving the decimal point one place to the left divides a number by 10. Likewise, moving a decimal point two places to the right multiplies the value of a number by 100, whereas moving the decimal point two places to the left divides a number by 100.

Example: 24.35 3 10 5 243.5 (decimal point moved to right) 24.35 4 10 5 2.435 (decimal point moved to left)

104. To change a fraction to a decimal, divide the numerator of the fraction by its denominator.

Example: Express 5 __ 6 as a decimal. We divide 5 by 6, obtaining 0.83.

5 __ 6 5 5 4 6 5 0.833…

105. To convert a decimal to a fraction, delete the decimal point and divide by whatever unit of 10 the number of decimal places represents.

Example: Convert 0.83 to a fraction. First, delete the decimal point. Second, two decimal

places represent hundredths, so divide 83 by 100: 83 ____ 100 .

0.83 5 83 ____ 100

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Complete sAt mAtH reFresHer • 175COMPLETE SAT MATH REFRESHER – session 1 • 175

106. To change a fraction to a percent, find its decimal form, multiply by 100, and add a percent sign.

Example: Express 3 __ 8 as a percent. To convert 3 __ 8 to a decimal, divide 3 by 8, which gives

us 0.375. Multiplying 0.375 by 100 gives us 37.5%.

107. To change a percent to a fraction, drop the percent sign and divide the number by 100.

Example: Express 17% as a fraction. Dropping the % sign gives us 17, and dividing by

100 gives us 17 ____ 100 .

108. To reduce a fraction, divide the numerator and denominator by the largest number that divides them both evenly.

Example: Reduce 10 ___ 15 . Dividing both the numerator and denominator by 5 gives us 2 __ 3 .

Example: Reduce 12 ___ 36 . The largest number that divides into both 12 and 36 is 12. Reducing

the fraction, we have 1 12

___ 36 3

5 1 __ 3 .

Note: In both examples, the reduced fraction is exactly equal to the original fraction:

2 __ 3 5 10 ___ 15 and 12 ___ 36 5 1 __ 3 .

109. To add fractions with like denominators, add the numerators of the fractions, keeping the same denominator.

Example: 1 __ 7 + 2 __ 7 + 3 __ 7 5 6 __ 7 .

110. To add fractions with different denominators, you must first change all of the fractions to equivalent fractions with the same denominators.

StEp 1. Find the lowest (or least) common denominator, the smallest number divisible by all of the denominators.

Example: If the fractions to be added are 1 __ 3 , 1 __ 4 , and 5 __ 6 , then the lowest common denomi-

nator is 12, because 12 is the smallest number that is divisible by 3, 4, and 6.

StEp 2. Convert all of the fractions to equivalent fractions, each having the lowest common denominator as its denominator. To do this, multiply the numerator of each fraction by the number of times that its denominator goes into the lowest common denominator. The product of this multiplication will be the new numerator. The denominator of the equivalent fractions will be the lowest common denominator. (See Step 1 above.)

Example: The lowest common denominator of 1 __ 3 , 1 __ 4 , and 5 __ 6 is 12. Thus, 1 __ 3 5 4 ___ 12 , because

12 divided by 3 is 4, and 4 times 15 4. 1 __ 4 5 3 ___ 12 , because 12 divided by 4 is 3, and 3 times

1 5 3. 5 __ 6 5 10 ___ 12 , because 12 divided by 6 is 2, and 2 times 5 5 10.

StEp 3. Now add all of the equivalent fractions by adding the numerators.

Example: 4 ___ 12 1 3 ___ 12 1 10 ___ 12 5 17 ___ 12

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176 • Gruber’s Complete sAt Guide 2015

StEp 4. Reduce the fraction if possible, as shown in Section 108.

Example: Add 4 __ 5 , 2 __ 3 , and 8 ___ 15 . The lowest common denominator is 15, because 15 is the

smallest number that is divisible by 5, 3, and 15. Then, 4 __ 5 is equivalent to 12 ___ 15 ; 2 __ 3 is equivalent

to 10 ___ 15 ; and 8 ___ 15 remains as 8 ___ 15 . Adding these numbers gives us 12 ___ 15 1 10 ___ 15 1 8 ___ 15 5 30 ___ 15 . Both 30

and 15 are divisible by 15, giving us 2 __ 1 , or 2.

111. To multiply fractions, follow this procedure:

StEp 1. To find the numerator of the product, multiply all the numerators of the fractions being multiplied.

StEp 2. To find the denominator of the product, multiply all of the denominators of the fractions being multiplied.

StEp 3. Reduce the product.

Example: 5 __ 7 × 2 ___ 15

5 5 __ 7 × 2 ___ 15 5 2 ___ 21 . We reduced by dividing both the numerator and

denom inator by 5, the common factor.

112. To divide fractions, follow this procedure:

StEp 1. Invert the divisor. That is, switch the positions of the numerator and denominator in the fraction you are dividing by.

StEp 2. Replace the division sign with a multiplication sign.

StEp 3. Carry out the multiplication indicated.

StEp 4. Reduce the product.

Example: Find 3 __ 4 ÷ 7 __ 8 . Inverting 7 __ 8 , the divisor, gives us 8 __ 7 . Replacing the division sign

with a multiplication sign gives us 3 __ 4 × 8 __ 7 . Carrying out the multiplication gives us 3 __ 4 × 8 __ 7

5 24 ___ 28

. The fraction 24 ___ 28 may then be reduced to 6 __ 7 by dividing both the numerator and the

denominator by 4.

113. To multiply decimals, follow this procedure:

StEp 1. Disregard the decimal point. Multiply the factors (the numbers being multiplied) as if they were whole numbers.

StEp 2. In each factor, count the number of digits to the right of the decimal point. Find the total number of these digits in all the factors. In the product, start at the right and count to the left this (total) number of places. Put the decimal point there.

Example: Multiply 3.8 3 4.01. First, multiply 38 and 401, getting 15,238. There is a total of 3 digits to the right of the decimal points in the factors. Therefore, the decimal point in the product is placed 3 units to the left of the digit farthest to the right (8).

3.8 3 4.01 5 15.238

Example: 0.025 3 3.6. First, multiply 25 3 36, getting 900. In the factors, there is a total of 4 digits to the right of the decimal points; therefore, in the product, we place the decimal point 4 units to the left of the digit farthest to the right in 900. However, there are only 3 digits in the product, so we add a 0 to the left of the 9, getting 0900. This makes it possible to place the decimal point correctly, thus: .0900, or .09. From this example, we can derive the rule that in the product we add as many zeros as are needed to provide the proper number of digits to the left of the digit farthest to the right.

1

3

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114. To find a percent of a given quantity:

StEp 1. Replace the word “of ” with a multiplication sign.

StEp 2. Convert the percent to a decimal: drop the percent sign and divide the number by 100. This is done by moving the decimal point two places to the left, adding zeros where necessary.

Examples: 30% 5 0.30 2.1% 5 0.021 78% 5 0.78

StEp 3. Multiply the given quantity by the decimal.

Example: Find 30% of 200.

30% of 200 5 30% 3 200 5 0.30 3 200 5 60.00

deviationsEstimation problems arise when dealing with approximations, that is, numbers that are not mathematically precise. The error, or deviation, in an approximation is a measure of the close-ness of that approximation.

115. Absolute error, or absolute deviation, is the difference between the estimated value and the real value (or between the approximate value and the exact value).

Example: If the actual value of a measurement is 60.2 and we estimate it as 60, then the absolute deviation (absolute error) is 60.2 2 60 5 0.2.

116. Fractional error, or fractional deviation, is the ratio of the absolute error to the exact value of the quantity being measured.

Example: If the exact value is 60.2 and the estimated value is 60, then the fractional error is

60.2 − 60 _________ 60.2 5 0.2 ____ 60.2 5 0.2 × 5 ________ 60.2 × 5 5 1 ____ 301 .

117. Percent error, or percent deviation, is the fractional error expressed as a percent. (See Section 106 on page 175 for the method of converting fractions to percents.)

118. Many business problems, including the calculation of loss, profit, interest, and so forth, are treated as deviation problems. Generally, these problems concern the difference between the original value of a quantity and some new value after taxes, after interest, etc. The following chart shows the relationship between business and estimation problems.

Business Problems Estimation Problemsoriginal value 5 exact value

new value 5 approximate value

net profit net loss } 5 absolute errornet interest

fractional profit fractional loss } 5 fractional errorfractional interest

percent profitpercent loss } 5 percent errorpercent interest

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Example: An item that originally cost $50 is resold for $56. Thus the net profit is

$56 2 $50 5 $6. The fractional profit is $56 − $50 _________ $50

5 $6 ____ $50

5 3 ___ 25

. The percent profit is

equal to the percent equivalent of 3 ___ 25 , which is 12%. (See Section 106 for converting fractions to percents.)

119. When there are two or more consecutive changes in value, remember that the new value of the first change becomes the original value of the second; consequently, successive fractional or percent changes may not be added directly.

Example: Suppose that a $100 item is reduced by 10% and then by 20%. The first reduction puts the price at $90 (10% of $100 5 $10; $100 2 $10 5 $90). Then, reducing the $90 (the new original value) by 20% gives us $72 (20% of $90 5 $18; $90 2 $18 5 $72). Therefore, it is not correct to simply add 10% and 20% and then take 30% of $100.

Ratios and Proportions120. A proportion is an equation stating that two ratios are equal. For example, 3 : 2 5 9 : x and 7 : 4 5 a : 15 are proportions. To solve a proportion:

StEp 1. First change the ratios to fractions. To do this, remember that a : b is the same

as a __ b , or 1 : 2 is equivalent to 1 __ 2 , or 7 : 4 5 a : 15 is the same as 7 __ 4 5 a ___

15 .

StEp 2. Now cross-multiply. That is, multiply the numerator of the first fraction by the denominator of the second fraction. Also multiply the denominator of the first fraction by the numerator of the second fraction. Set the first product equal to the second. This rule is sometimes stated as “The product of the means equals the product of the extremes.”

Example: When cross-multiplying in the equation 3 __ 2 5 9 __ y , we get 3 3 y 5 2 3 9, or 3y 5 18. Dividing by 3, we get y 5 6.

When we cross-multiply in the equation a __ 2 5 4 __ 8 , we get 8a 5 8, and by dividing each side

of the equation by 8 to reduce, a 5 1.

StEp 3. Solve the resulting equation. This is done algebraically.

Example: Solve for a in the proportion 7 : a 5 6 : 18.

Change the ratios to the fractional relation 7 __ a 5 6 ___ 18

. Cross-multiply: 7 3 18 5 6 3 a, or 126 5 6a.

Solving for a gives us a 5 21.

121. In solving proportions that have units of measurement (feet, seconds, miles, etc.), each ratio must have the same units. For example, if we have the ratio 5 inches : 3 feet, we must convert the 3 feet to 36 inches and then set up the ratio 5 inches : 36 inches, or 5 : 36. We might

wish to convert inches to feet. Noting that 1 inch 5 1 ___ 12 foot, we get 5 inches : 3 feet 5 5 ( 1 ___ 12 ) feet: 3 feet 5 5 ___ 12 feet : 3 feet.

Example: On a blueprint, a rectangle measures 6 inches in width and 9 inches in length. If the actual width of the rectangle is 16 inches, how many feet are there in the length?

Solution: We set up the proportions, 6 inches : 9 inches 5 16 inches : x feet. Since x feet is equal to 12x inches, we substitute this value in the proportion. Thus, 6 inches : 9 inches 5 16 inches : 12x inches. Since all of the units are now the same, we may work

with the numbers alone. In fractional terms we have 6 __ 9 5 16 ____ 12x

. Cross-multiplication

gives us 72x 5 144, and solving for x gives us x 5 2. The rectangle is 2 feet long.

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Variations122. In a variation problem, you are given a relationship between certain variables. The problem is to determine the change in one variable when one or more of the other variables changes.

Direct Variation (Direct Proportion)If x varies directly with y, this means that x __ y 5 k (or x 5 ky) where k is a constant.

Example: If the cost of a piece of glass varies directly with the area of the glass, and a piece of glass of 5 square feet costs $20, then how much does a piece of glass of 15 square feet cost?

Represent the cost of the glass as c and the area of the piece of glass as A. Then we have

c __ A

5 k.

Now since we are given that a piece of glass of 5 square feet costs $20, we can write 20 ___ 5 5 k,

and we find k 5 4.

Let’s say a piece of glass of 15 square feet costs $ x. Then we can write x ___ 15 5 k. But we found

k 5 4, so x ___ 15 5 4 and x 5 60. $60 is then the answer.

Inverse Variation (Inverse Proportion)If x varies inversely with y, this means that xy 5 k where k is a constant.

Example: If a varies inversely with b, and when a 5 5, b 5 6, then what is b when a 5 10?

We have ab 5 k. Since a 5 5 and b 5 6, 5 3 6 5 k 5 30. So if a 5 10, 10 3 b 5 k 5 30 and b 5 3.

Other Variations

Example: In the formula A 5 bh, if b doubles and h triples, what happens to the value of A?

StEp 1. Express the new values of the variables in terms of their original values, that is, b' 5 2b and h' 5 3h.

StEp 2. Substitute these values in the formula and solve for the desired variable: A' 5 b' h' 5 (2b)(3h) 5 6bh.

StEp 3. Express this answer in terms of the original value of the variable, that is, since the new value of A is 6bh, and the old value of A was bh, we can express this as Anew 5 6Aold. The new value of the variable is expressed with a prime mark and the old value of the variable is left as it was. In this problem, the new value of A would be expressed as A' and the old value as A. A' 5 6A.

Example: If V 5 e3 and e is doubled, what happens to the value of V?

Solution: Replace e with 2e. The new value of V is (2e)3. Since this is a new value, V becomes V'. Thus V' 5 (2e)3, or 8e3. Remember, from the original statement of the problem, that V 5 e3. Using this, we may substitute V for e3 found in the equation V' 5 8e3. The new equation is V' 5 8V. Therefore, the new value of V is 8 times the old value.

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Comparison of FractionsIn fraction comparison problems, you are given two or more fractions and are asked to arrange them in increasing or decreasing order, or to select the larger or the smaller. The following rules and suggestions will be very helpful in determining which of two fractions is greater.

123. If fractions A and B have the same denominators, and A has a larger numerator, then fraction A is larger. (We are assuming here, and for the rest of this Refresher Session, that numerators and denominators are positive.)

Example: 56 ____ 271 is greater than 53 ____ 271 because the numerator of the first fraction is greater

than the numerator of the second.

124. If fractions A and B have the same numerator, and A has a larger denominator, then fraction A is smaller.

Example: 37 ____ 256 is smaller than 37 ____ 254 .

125. If fraction A has a larger numerator and a smaller denominator than fraction B, then fraction A is larger than B.

Example: 6 ___ 11 is larger than 4 ___ 13 . (If this does not seem obvious, compare both fractions

with 6 ___ 13 .)

126. Another method is to convert all of the fractions to equivalent fractions. To do this follow these steps:

StEp 1. First find the lowest common denominator of the fractions. This is the smallest number that is divisible by all of the denominators of the original fractions. See Section 110 for the method of finding lowest common denominators.

StEp 2. The fraction with the greatest numerator is the largest fraction.

127. Still another method is the conversion to approximating decimals.

Example: To compare 5 __ 9 and 7 ___ 11 , we might express both as decimals to a few places of

accuracy: 5 __ 9 is approximately equal to 0.555, while 7 ___ 11 is approximately equal to 0.636,

so 7 ___ 11 is obviously greater. To express a fraction as a decimal, divide the numerator by

the denominator.

128. If all of the fractions being compared are very close in value to some easy-to-work-with

number, such as 1 __ 2 or 5, you may subtract this number from each of the fractions without

changing this order.

Example: To compare 151 ____ 75 with 328 ____ 163 , we notice that both of these fractions are approxi-

mately equal to 2. If we subtract 2 (that is, 150 ____ 75 and 326 ____ 163 , respectively) from each, we

get 1 ___ 75 and 2 ____ 163 , respectively. Since 1 ___ 75 (or 2 ____ 150 ) exceeds 2 ____ 163 , we see that 151 ____ 75 must also

exceed 328 ____ 163 .

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An alternative method of comparing fractions is to change the fractions to their decimal equivalents and then compare the decimals. (See Sections 104 and 127.) You should weigh the relative amount of work and difficulty involved in each method when you face each problem.

129. The following is a quick way of comparing fractions.

Example: Which is greater, 3 __ 8 or 7 ___ 18 ?

Procedure:

3 __ 8 7 ___ 18 MULtIpLY MULtIpLY

Multiply the 18 by the 3. We get 54. Put the 54 on the left side.

54

Now multiply the 8 by the 7. We get 56. Put the 56 on the right side.

54 56

Since 56 . 54 and 56 is on the right side, the fraction 7 ___ 18 (which was also originally on the

right side) is greater than the fraction 3 __ 8 (which was originally on the left side).

Example: If y . x, which is greater, 1 __ x or 1 __ y ? (x and y are positive numbers).

Procedure:

1 __ x 1 __ y MULtIpLY MULtIpLY

Multiply y by 1. We get y. Put y on the left side:

y

Multiply x by 1. We get x. Put x on the right side:

y x

Since y . x (given), 1 __ x (which was originally on the left) is greater than 1 __ y (which was originally on the right).

Example: Which is greater?

7 __ 9 or 3 __ 4

7 __ 9 3 __ 4

$ $28 . 27 $ $ 7 __ 9 . 3 __ 4

MULtIpLY

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practice test 1Fractions, Decimals, Percentages, Deviations, Ratios and Proportions, Variations, and Comparison of Fractions

Correct answers and solutions follow each test.

1. Which of the following answers is the sum of the following numbers:

2 1 __ 2 , 21 ___ 4 , 3.350, 1 __ 8 ?

(A) 8.225(B) 9.825(C) 10.825(D) 11.225(E) 12.350

2. A chemist was preparing a solution that should have included 35 milligrams of a chemical. If she actually used 36.4 milligrams, what was her percentage error (to the nearest 0.01%)?

(A) 0.04%(B) 0.05%(C) 1.40%(D) 3.85%(E) 4.00%

3. A retailer buys a popular brand of athletic shoe from the wholesaler for $75. He then marks

up the price by 1 __ 3 and sells each pair at a discount of 20%. What profit does the retailer make

on each pair of athletic shoes?

(A) $5.00(B) $6.67(C) $7.50(D) $10.00(E) $13.33

4. On a blueprint, 1 __ 4 inch represents 1 foot. If a window is supposed to be 56 inches wide, how

wide would its representation be on the blueprint?

(A) 1 1 __ 6 inches

(B) 4 2 __ 3 inches

(C) 9 1 __ 3 inches

(D) 14 inches

(E) 18 2 __ 3 inches

5. If the radius of a circle is increased by 50%, what will be the percent increase in the circumfer-ence of the circle? (Circumference 5 2π r)

(A) 25%(B) 50%(C) 100%(D) 150%(E) 225%

A B C D E

1.

A B C D E

2.

A B C D E

3.

A B C D E

4.

A B C D E

5.

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6. Which of the following fractions is the greatest?

(A) 403 ____ 134

(B) 79 ___ 26

(C) 527 ____ 176

(D) 221 ____ 73

(E) 99 ___ 34

7. A store usually sells a certain item at a 40% profit. One week the store has a sale, during which the item is sold for 10% less than the usual price. During the sale, what is the percent profit the store makes on each of these items?

(A) 4%(B) 14%(C) 26%(D) 30%(E) 36%

8. What is 0.05 percent of 6.5?

(A) 0.00325(B) 0.013(C) 0.325(D) 1.30(E) 130.0

9. What is the value of ( 3 1 __ 2 + 3 1 __ 4 + 3 1 __ 4 + 3 1 __ 2 )

___________________ 4 1 __ 2

?

(A) 1 1 __ 2

(B) 2 1 __ 4

(C) 3

(D) 3 1 __ 4

(E) 3 3 __ 8

10. If 8 loggers can chop down 28 trees in one day, how many trees can 20 loggers chop down in one day?

(A) 28 trees(B) 160 trees(C) 70 trees(D) 100 trees(E) 80 trees

A B C D E

6.

A B C D E

7.

A B C D E

8.

A B C D E

9.

A B C D E

10.

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11. What is the product of the following fractions: 3 ____ 100 , 15 ___ 49 , 7 __ 9 ?

(A) 215 ______ 44,100

(B) 1 ____ 140

(C) 1 ____ 196

(D) 25 ____ 158

(E) 3 ____ 427

12. In calculating the height of an object, Mrs. Downs mistakenly observed the height to be 72 cm instead of 77 cm. What was her percentage error (to the nearest hundredth of a percent)?

(A) 6.49%(B) 6.69%(C) 6.89%(D) 7.09%(E) 7.19%

13. A retailer buys 1,440 dozen pens at $2.50 a dozen and then sells them at a price of 25¢ apiece. What is the total profit after the retailer sells all the pens?

(A) $60.00(B) $72.00(C) $720.00(D) $874.00(E) $8,740.00

14. On a map, 1 inch represents 1,000 miles. If the area of a country is actually 16 million square miles, what is the area of the country’s representation on the map?

(A) 4 square inches(B) 16 square inches(C) 4,000 square inches(D) 16,000 square inches(E) 4,000,000 square inches

15. The formula for the volume of a cone is V 5 1 __ 3 π r 2h. If the radius (r) is doubled and the height

(h) is divided by 3, what will be the ratio of the new volume to the original volume?

(A) 2 : 3(B) 3 : 2(C) 4 : 3(D) 3 : 4(E) None of these.

16. Which of the following fractions has the smallest value?

(A) 34.7 ____ 163

(B) 125 ____ 501

(C) 173 ____ 700

(D) 10.9 ____ 42.7

(E) 907 _____ 3,715

A B C D E

11.

A B C D E

12.

A B C D E

13.

A B C D E

14.

A B C D E

15.

A B C D E

16.

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17. Mr. Cutler usually makes a 45% profit on every flat-screen TV he sells. During a sale, he reduces his margin of profit to 40%, while his sales increase by 10%. What is the ratio of his new total profit to the original profit?

(A) 1 : 1(B) 9 : 8(C) 9 : 10(D) 11 : 10(E) 44 : 45

18. What is 1.3 percent of 0.26?

(A) 0.00338(B) 0.00500(C) 0.200(D) 0.338(E) 0.500

19. What is the average of the following numbers: 3.2, 47 ___ 12 , 10 ___ 3 ?

(A) 3.55

(B) 10 ___ 3

(C) 103 ____ 30

(D) 209 ____ 60

(E) 1,254 _____ 120

20. If it takes 16 faucets 10 hours to fill 8 tubs, how long will it take 12 faucets to fill 9 tubs?

(A) 10 hours(B) 12 hours(C) 13 hours(D) 14 hours(E) 15 hours

21. If the 8% tax on a sale amounts to 96¢, what is the final price (tax included) of the item?

(A) $1.20(B) $2.16(C) $6.36(D) $12.00(E) $12.96

22. In a certain class, 40% of the students are girls, and 20% of the girls wear glasses. What per-cent of the children in the class are girls who wear glasses?

(A) 6%(B) 8%(C) 20%(D) 60%(E) 80%

23. What is 1.2% of 0.5?

(A) 0.0006(B) 0.006(C) 0.06(D) 0.6(E) 6.0

A B C D E

17.

A B C D E

18.

A B C D E

19.

A B C D E

20.

A B C D E

21.

A B C D E

22.

A B C D E

23.

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24. Which of the following quantities is the largest?

(A) 275 ____ 369

(B) 134 ____ 179

(C) 107 ____ 144

(D) 355 ____ 476

(E) 265 ____ 352

25. If the length of a rectangle is increased by 120%, and its width is decreased by 20%, what hap-pens to the area of the rectangle?

(A) It decreases by 4%.(B) It remains the same.(C) It increases by 24%.(D) It increases by 76%.(E) It increases by 100%.

26. A merchant buys an old carpet for $25.00. He spends $15.00 to have it restored to good condi-tion and then sells the rug for $50.00. What is the percent profit on his total investment?

(A) 20%(B) 25%(C) 40%

(D) 66 2 __ 3 %

(E) 100%

27. Of the following sets of fractions, which one is arranged in decreasing order?

(A) 5 __ 9 , 7 ___ 11 , 3 __ 5 , 2 __ 3 , 10 ___ 13

(B) 2 __ 3 , 3 __ 5 , 7 ___ 11 , 5 __ 9 , 10 ___ 13

(C) 3 __ 5 , 5 __ 9 , 7 ___ 11 , 10 ___ 13 , 2 __ 3

(D) 10 ___ 13 , 2 __ 3 , 7 ___ 11 , 3 __ 5 , 5 __ 9

(E) None of these.

28. If the diameter of a circle doubles, the circumference of the larger circle is how many times the circumference of the original circle? (Circumference 5π d)

(A) π(B) 2π(C) 1(D) 2(E) 4

29. The scale on a set of plans is 1 : 8. If a man reads a certain measurement on the plans as 5.60, instead of 6.00, what will be the resulting approximate percent error on the full-size model?

(A) 6.7%(B) 7.1%(C) 12.5%(D) 53.6%(E) 56.8%

A B C D E

24.

A B C D E

25.

A B C D E

26.

A B C D E

27.

A B C D E

28.

A B C D E

29.

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30. G&R Electronics bought 2 dozen megapixel digital cameras for $300 each. The company sold two-thirds of them at a 25% profit but was forced to take a 30% loss on the rest. What was the total profit (or loss) on the digital cameras?

(A) a loss of $200(B) a loss of $15(C) no profit or loss(D) a profit of $20(E) a profit of $480

31. The sum of 1 __ 2 , 1 __ 3 , 1 __ 8 , and 1 ___ 15 is:

(A) 9 __ 8

(B) 16 ___ 15

(C) 41 ___ 40

(D) 65 ___ 64

(E) 121 ____ 120

32. What is 2 __ 3 % of 90?

(A) 0.006(B) 0.06(C) 0.6(D) 6.0(E) 60

33. Lucas borrows $360. If he pays it back in 12 monthly installments of $31.50, what is the inter-est rate?

(A) 1.5%(B) 4.5%(C) 10%(D) 5%(E) 7.5%

34. A merchant marks up a certain lighting fixture 30% above original cost. Then the merchant gives a customer a loyalty discount of 15%. If the final selling price for the lighting fixture was $86.19, what was the original cost?

(A) $66.30(B) $73.26(C) $78.00(D) $99.12(E) $101.40

35. In a certain recipe, 2 1 __ 4 cups of flour are called for to make a cake that serves 6. If Mrs. Jenkins

wants to use the same recipe to make a cake for 8, how many cups of flour must she use?

(A) 2 1 __ 3 cups

(B) 2 3 __ 4 cups

(C) 3 cups

(D) 3 3 __ 8 cups

(E) 4 cups

A B C D E

30.

A B C D E

31.

A B C D E

32.

A B C D E

33.

A B C D E

34.

A B C D E

35.

SAT2015_P06.indd 187 4/23/14 11:39 AM


36. If 10 people can survive for 24 days on 15 cans of rations, how many cans will be needed for 8 people to survive for 36 days?

(A) 15 cans(B) 16 cans(C) 17 cans(D) 18 cans(E) 19 cans

37. If, on a map, 1 __ 2 inch represents 1 mile, how long is a border whose representation is 1 1 ___ 15 feet long?

(A) 2 1 ___ 30 miles

(B) 5 1 ___ 15 miles

(C) 12 4 __ 5 miles

(D) 25 3 __ 5 miles

(E) 51 1 __ 5 miles

38. In the formula e 5 hf, if e is doubled and f is halved, what happens to the value of h?

(A) h remains the same.(B) h is doubled.(C) h is divided by 4.(D) h is multiplied by 4.(E) h is halved.

39. Which of the following expresses the ratio of 3 inches to 2 yards?

(A) 3 : 2(B) 3 : 9(C) 3 : 12(D) 3 : 24(E) 3 : 72

40. If it takes Mark twice as long to earn $6.00 as it takes Carl to earn $4.00, what is the ratio of Mark’s pay per hour to Carl’s pay per hour?

(A) 2 : 1(B) 3 : 1(C) 3 : 2(D) 3 : 4(E) 4 : 3

41. What is the lowest common denominator of the following set of fractions:

1 __ 6 , 13 ___ 27 , 4 __ 5 , 3 ___ 10 , 2 ___ 15 ?

(A) 27(B) 54(C) 135(D) 270(E) None of these.

A B C D E

36.

A B C D E

37.

A B C D E

38.

A B C D E

39.

A B C D E

40.

A B C D E

41.

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42. The average grade on a certain examination was 85. Raul scored 90 on the same examination. What was Raul’s percent deviation from the average score (to the nearest tenth of a percent)?

(A) 5.0%(B) 5.4%(C) 5.5%(D) 5.8%(E) 5.9%

43. Successive discounts of 20% and 12% are equivalent to a single discount of:

(A) 16.0%(B) 29.6%(C) 31.4%(D) 32.0%(E) 33.7%

44. On a blueprint of a park, 1 foot represents 1 __ 2 mile. If an error of 1 __ 2 inch is made in reading the

blueprint, what will be the corresponding error on the actual park?

(A) 110 feet(B) 220 feet(C) 330 feet(D) 440 feet(E) None of these.

45. If the two sides of a rectangle change in such a manner that the rectangle’s area remains constant, and one side increases by 25%, what must happen to the other side?

(A) It decreases by 20%(B) It decreases by 25%

(C) It decreases by 33 1 __ 3 %

(D) It decreases by 50%(E) None of these.

46. Which of the following fractions has the smallest value?

(A) 6,043 _____ 2,071

(B) 4,290 _____ 1,463

(C) 5,107 _____ 1,772

(D) 8,935 _____ 2,963

(E) 8,016 _____ 2,631

47. A certain company increased its prices by 30% during 2011. Then, in 2012, it wasforced to cut back its prices by 20%. What was the net change in price?

(A) 24%(B) 22%(C) 12%(D) 14%(E) 0%

A B C D E

42.

A B C D E

43.

A B C D E

44.

A B C D E

45.

A B C D E

46.

A B C D E

47.

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48. What is 0.04%, expressed as a fraction?

(A) 2 __ 5

(B) 1 ___ 25

(C) 4 ___ 25

(D) 1 ____ 250

(E) 1 _____ 2,500

49. What is the value of the fraction 16 + 12 + 88 + 34 + 66 + 21 + 79 + 11 + 89 ______________________________________ 25 ?

(A) 15.04(B) 15.44(C) 16.24(D) 16.64(E) None of these.

50. If coconuts are twice as expensive as bananas, and bananas are one-third as expensive as grapefruits, what is the ratio of the price of one coconut to one grapefruit?

(A) 2 : 3(B) 3 : 2(C) 6 : 1(D) 1 : 6(E) None of these.

A B C D E

48.

A B C D E

49.

A B C D E

50.

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Answer Key for practice test 1 1. D 14. B 27. D 39. E 2. E 15. C 28. D 40. D 3. A 16. A 29. A 41. D 4. A 17. E 30. E 42. E 5. B 18. A 31. C 43. B 6. B 19. D 32. C 44. A 7. C 20. E 33. D 45. A 8. A 21. E 34. C 46. C 9. C 22. B 35. C 47. D 10. C 23. B 36. D 48. E 11. B 24. E 37. D 49. D 12. A 25. D 38. D 50. A 13. C 26. B

Answers and solutions for practice test 1

1. Choice D is correct. First, convert the fractions to decimals, as the final answer must be expressed in decimals: 2.500 1 5.250 1 3.350 1 0.125 5 11.225.

(104, 127, 128)

2. Choice E is correct. This is an estimation problem. Note that the correct value was 35, not 36.4. Thus the real value is 35 mg and the estimated value is 36.4 mg. Thus, percent error is equal to (36.4 2 35) 4 35, or 0.04, expressed as a percent, which is 4%.

(115, 116, 117)

3. Choice A is correct. This is a business problem.

First, the retailer marks up the wholesale price by 1 __ 3 ,

so the marked-up price equals $75(1 1 1 __ 3 ), or $100;

then it is reduced 20% from the $100 price, leaving a final price of $80. Thus, the net profit on each pair of athletic shoes is $5.00. (118)

4. Choice A is correct. Here we have a proportion

problem: length on blueprint : actual length 5 1 __ 4

inch : 1 foot. The second ratio is the same as 1 : 48, because 1 foot 5 12 inches. In the problem the actual length is 56 inches, so that if the length on the blueprint equals x, we have the proportion

x : 56 5 1 : 48; x ___ 56 5 1 ___ 48

. 48x 5 56; so x 5 56 ___ 48 , or

1 1 __ 6 inches. (120)

5. Choice B is correct. C 5 2π r (where r is the radius of the circle, and C is its circumference). The new value of r, r', is (1.5)r since r is increased by 50%. Using this value of r', we get the new C, C' 5 2π r' 5 2π(1.5)r 5 (1.5)2π r. Remembering that C 5 2π r, we get that C' 5 (1.5)C. Since the new

circumference is 1.5 times the original, there is an increase of 50%. (122)

6. Choice B is correct. In this numerical comparison problem, it is helpful to realize that all of these frac-tions are approximately equal to 3. If we subtract

3 from each of the fractions, we get 1 ____ 134 , 1 ___ 26 ,

− 1 ____ 176

, 2 ___ 73 , and − 3 ___ 34

, respectively. Clearly, the

greatest of these is 1 ___ 26 , which therefore shows the

greatest of the five given fractions. Another method of solving this type of numerical comparison problem is to convert the fractions to decimals by dividing the numerator by the denominator. (127, 128)

7. Choice C is correct. This is another business problem, this time asking for percentage profit. Let the original price be P. A 40% profit means that the store will sell the item for 100%P 1 40%P, which is equal to 140%P, which in turn is equal to

( 140 ____ 100 ) P 5 1.4P. Then the marked-up price will be

1.4(P). Ten percent is taken off this price, to yield a final price of (0.90)(1.40)(P), or (1.26)(P). Thus, the fractional increase was 0.26, so the percent increase was 26%. (118)

8. Choice A is correct. Remember that the phrase “percent of ” may be replaced by a multiplication sign. Thus, 0.05% 3 6.5 5 0.0005 3 6.5, so the answer is 0.00325. (114)

9. Choice C is correct. First, add the fractions in the

numerator to obtain 13 1 __ 2 . Then divide 13 1 __ 2 by 4 1 __ 2 . If

you cannot see immediately that the answer is 3, you can convert the halves to decimals and divide, or you can express the fractions in terms of their

common denominator, thus: 13 1 __ 2 5 27 ___ 2 ; 4 1 __ 2 5 9 __ 2 ;

27 ___ 2 4 9 __ 2 5 27 ___ 2 3 2 __ 9 5 54 ___ 18 5 3. (110, 112)

10. Choice C is correct. This is a proportion problem. If x is the number of loggers needed to chop down 20 trees, then we form the proportion 8 loggers : 28

trees 5 20 loggers : x trees, or 8 ___ 28 5 20 ___ x . Solving for

x, we get x 5 (28)(20) ________ 8 , or x 5 70. (120)

11. Choice B is correct. 3 ____ 100 × 15 ___ 49

× 7 __ 9 5 3 × 15 × 7 ____________

100 × 49 × 9 .

Canceling 7 out of the numerator and denominator

gives us 3 × 15 ___________ 100 × 7 × 9 . Canceling 5 out of the numer-

ator and denominator gives us 3 × 3 __________ 20 × 7 × 9 . Finally,

canceling 9 out of both numerator and denominator

gives us 1 _______ 20 × 7 , or 1 ____ 140 . (111)

SAT2015_P06.indd 191 4/23/14 11:39 AM


12. Choice A is correct. Percent error 5 (absolute error) 4 (correct measurement) 5 5 4 77 5 0.0649 (approximately) 3 100 5 6.49%. (115, 116, 117)

13. Choice C is correct. Profit on each dozen pens 5 selling price 2 cost 5 12(25¢) 2 $2.50 5 $3.00 2

$2.50 5 50¢ profit per dozen. Total profit 5 profit per dozen 3 number of dozens 5 50¢ 3 1440 5 $720.00. (118)

14. Choice B is correct. If 1 inch represents 1,000 miles, then 1 square inch represents 1,000 miles squared, or 1,000,000 square miles. Thus, the area would be represented by 16 squares of this size, or 16 square inches. (120)

15. Choice C is correct. Let V' equal the new volume.

Then if r' 5 2r is the new radius, and h' 5 h __ 3 is

the new height, V' 5 1 __ 3 π(r')2(h') 5 1 __ 3 π(2r)2 ( h __ 3 ) 5

4 __ 9 π r 2h 5 4 __ 3 V, so the ratio V' : V is equal to 4 : 3. (122)

16. Choice A is correct. Using a calculator, we get: 34.7 ____ 163

5 0.2128 for Choice A; 125 ____ 501 5 0.2495 for Choice

B; 173 ____ 700 5 0.2471 for Choice C; 10.9 ____ 42.7 5 0.2552 for

Choice D; and 907 _____ 3,715 5 0.2441 for Choice E. Choice

A is the smallest value. (104, 127)

17. Choice E is correct. Let N 5 the original cost of a flat-screen TV. Then, original profit 5 45% 3 N. New profit 5 40% 3 110%N 5 44% 3 N. Thus, the ratio of new profit to original profit is 44 : 45. (118)

18. Choice A is correct.1.3% 3 0.26 5 0.013 3 0.26 5 0.00338. (114)

19. Choice D is correct. Average 5 1 __ 3 ( 3.2 + 47 ___ 12 + 10 ___ 3 ) . The decimal 3.2 5 320 ____

100 5 16 ___ 5 , and the lowest

common denominator of the three fractions is 60,

then 16 ___ 5 5 192 ____ 60

, 47 ___ 12 5 235 ____ 60

, and 10 ___ 3 5 200 ____ 60

. Then,

1 __ 3 ( 192 ____ 60 + 235 ____ 60 + 200 ____ 60 ) 5 1 __ 3 ( 627 ____ 60 ) 5 209 ____

60 .

(101, 105, 109)

20. Choice E is correct. This is an inverse proportion. If it takes 16 faucets 10 hours to fill 8 tubs, then it takes 1 faucet 160 hours to fill 8 tubs (16 faucets :

1 faucet 5 x hours : 10 hours; 16 ___ 1 5 x ___ 10

; x 5 160). If

it takes 1 faucet 160 hours to fill 8 tubs, then (divid-ing by 8) it takes 1 faucet 20 hours to fill 1 tub. If it takes 1 faucet 20 hours to fill 1 tub, then it takes 1 faucet 180 hours (9 3 20 hours) to fill 9 tubs. If it

takes 1 faucet 180 hours to fill 9 tubs, then it takes

12 faucets 180 ____ 12 , or 15 hours to fill 9 tubs. (120)

21. Choice E is correct. Let P be the original price. Then 0.08P 5 96¢, so that 8P 5 $96, or P 5 $12. Adding the tax, which equals 96¢, we obtain our final price of $12.96. (118)

22. Choice B is correct. The number of girls who wear glasses is 20% of 40% of the children in the class. Thus, the indicated operation is multiplication; 20% 3 40% 5 0.20 3 0.40 5 0.08 5 8%. (114)

23. Choice B is correct. 1.2% 3 0.5 5 0.012 3 0.5 5 0.006. (114)

24. Choice E is correct. Using a calculator to find

the answer to three decimal places, we get: 275 ____ 369

5 0.745 for Choice A; 134 ____ 179 5 0.749 for Choice B;

107 ____ 144 5 0.743 for Choice C; 355 ____ 476 5 0.746 for Choice

D; 265 ____ 352 5 0.753 for Choice E. Choice E is the largest

value. (104, 127)

25. Choice D is correct. Area 5 length 3 width. The new area will be equal to the new length 3 the new width. The new length 5 (100% 1 120%) 3 old

length 5 220% 3 old length 5 220 ____ 100 3 old length 5

2.2 3 old length. The new width 5 (100% 2 20%)

3 old width 5 80% 3 old width 5 80 ____ 100 3 old width

5 .8 3 old width. The new area 5 new width 3 new length 5 2.2 3 .8 3 old length 3 old width. So the new area 5 1.76 3 old area, which is 176% of the old area. This is an increase of 76% from the original area. (122)

26. Choice B is correct. Total cost to merchant 5 $25.00 1 $15.00 5 $40.00.

Profit 5 selling price 2 cost 5 $50 2 $40 5 $10. Percent profit 5 profit 4 cost 5 $10 4 $40 5 25%.

(118)

27. Choice D is correct. We can convert the fractions to decimals or to fractions with a lowest common denominator. Inspection will show that all sets of fractions contain the same members; therefore, if we convert one set to decimals or find the low-est common denominator for one set, we can use our results for all sets. Converting a fraction to a decimal involves only one operation, a single divi-sion, whereas converting to the lowest common denominator involves a multiplication, which must be followed by a division and a multiplication to change each fraction to one with the lowest com-mon denominator. Thus, conversion to decimals is

often the simpler method: 10 ___ 13 5 0.769; 2 __ 3 5 0.666; 7 ___ 11

5 0.636; 3 __ 5 5 0.600; 5 __ 9 5 0.555.

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However, in this case there is an even simpler method. Convert two of the fractions to equivalent

fractions: 3 __ 5 5 6 ___ 10 and 2 __ 3 5 8 ___ 12 . We now have 5 __ 9 ,

6 ___ 10 , 7 ___ 11 , 8 ___ 12 , and 10 ___ 13 . Remember this rule: When the

numerator and denominator of a fraction are both

positive, adding 1 to both will bring the value of the

fraction closer to 1. (For example, 3 __ 4 5 2 + 1 _____ 3 + 1 , so 3 __ 4

is closer to 1 than 2 __ 3 and is therefore the greater

fraction.) Thus we see that 5 __ 9 is less than 6 ___ 10 , which

is less than 7 ___ 11 , which is less than 8 ___ 12 , which is less

than 9 ___ 13 . 9 ___ 13 is obviously less than 10 ___ 13 , so 10 ___ 13 must be

the greatest fraction. Thus, in decreasing order, the

fractions are 10 ___ 13 , 2 __ 3 , 7 ___ 11 , 3 __ 5 , and 5 __ 9 . This method is a

great time-saver once you become accustomed to it.

(104)

28. Choice D is correct. The formula governing this situation is C 5 π d, where C 5 circumference and d 5 diameter. Thus, if the new diameter is d' 5 2d, then the new circumference is C' 5 π d' 5 2π d 5 2C. Thus, the new, larger circle has a circumfer-ence twice that of the original circle. (122)

29. Choice A is correct. The most important feature of this problem is recognizing that the scale does not affect percent (or fractional) error, since it simply results in multiplying the numerator and denominator of a fraction by the same factor. Thus, we need only calculate the original percent error. Although it would not be incorrect to calculate the full-scale percent error, it would be time-consuming and might result in unnecessary errors. Absolute error 5 0.40. Actual measurement 5 6.00. Therefore, percent error 5 (absolute error 4

actual measurement) 3 100% 5 0.4 ___ 6.0 3 100%, which

equals 6.7% (approximately). (117)

30. Choice E is correct. Total cost 5 number of cameras 3 cost of each 5 24 3 $300 5 $7,200.

Revenue 5 (number sold at 25% profit 3 price at 25% profit) 1 (number sold at 30% loss 3 price at 30% loss)

5 (16 3 $375) 1 (8 3 $210) 5 $6,000 1 $1,680 5 $7,680.

Profit 5 revenue 2 cost 5 $7,680 2 $7,200 5 $480. (118)

31. Choice C is correct. 1 __ 2 1 1 __ 3 1 1 __ 8 1 1 ___ 15 5 60 ____ 120 1 40 ____ 120

1 15 ____ 120 1 8 ____ 120 5 123 ____ 120 5 41 ___ 40 . (110)

32. Choice C is correct. 2 __ 3 % 3 90 5 2 ____ 300 3 90 5 180 ____ 300

5 6 ___ 10 5 0.6. (114)

33. Choice D is correct. If Lucas makes 12 payments of $31.50, he pays back a total of $378.00. Since the loan is for $360.00, his net interest is $18.00.

Therefore, the rate of interest is $18.00 _______ $360.00

, which can

be reduced to 0.05, or 5%. (118)

34. Choice C is correct. Final selling price 5 85% 3 130% 3 cost 5 1.105 3 cost. Thus, $86.19 5 1.105C, where C 5 cost. C 5 $86.19 4 1.105 5 $78.00 (exactly). (118)

35. Choice C is correct. If x is the amount of flour needed for 8 people, then we can set up the propor-

tion 2 1 __ 4 cups : 6 people 5 x : 8 people. Solving for x

gives us x 5 8 __ 6 × 2 1 __ 4 or 8 __ 6 × 9 __ 4 5 3. (120)

36. Choice D is correct. If 10 people can survive for 24 days on 15 cans, then 1 person can survive for 240 days on 15 cans. If 1 person can survive for 240 days

on 15 cans, then 1 person can survive for 240 ____ 15 , or 16

days, on 1 can. If 1 person can survive for 16 days on

1 can, then 8 people can survive for 16 ___ 8 , or 2 days, on

1 can. If 8 people can survive for 2 days on 1 can, then

for 36 days 8 people need 36 ___ 2 , or 18 cans, to survive.

(120)

37. Choice D is correct. 1 1 ___ 15 feet 5 1 1 ___ 15 3 12 inches

5 16 ___ 15 3 12 inches 5 12.8 inches. So we have a pro-

portion, 1 __ 2 inch

______ 1 mile

5 12.8 inches __________ x miles

. Cross-multiplying,

we get 1 __ 2 x 5 12.8, so x 5 25.6 5 25 3 __ 5 . (120)

38. Choice D is correct. If e 5 hf, then h 5 e __ f . If e is

doubled and f is halved, then the new value of h, h'5

( 2e ___ 1 __ 2 f

) . Multiplying the numerator and denominator

by 2 gives us h' 5 4e ___ f . Since h 5 e __

f and h' 5 4e ___

f we

see that h' 5 4h. This is the same as saying that h is multiplied by 4. (122)

39. Choice E is correct. 3 inches : 2 yards 5 3 inches : 72 inches 5 3 : 72. (121)

40. Choice D is correct. If Carl and Mark work for the same length of time, then Carl will earn $8.00 for every $6.00 Mark earns (since in the time Mark can earn one $6.00 wage, Carl can earn two $4.00

SAT2015_P06.indd 193 4/23/14 11:39 AM


wages). Thus, their hourly wage rates are in the ratio $6.00 (Mark) : $8.00 (Carl) 5 3 : 4. (120)

41. Choice D is correct. The lowest common denomi-nator is the smallest number that is divisible by all of the denominators. Thus we are looking for the smallest number that is divisible by 6, 27, 5, 10, and 15. The smallest number that is divisible by 6 and 27 is 54. The smallest number that is divisible by 54 and 5 is 270. Since 270 is divisible by 10 and 15 also, it is the lowest common denominator. (110, 126)

42. Choice E is correct.

Percent deviation 5 absolute deviation ________________ average score × 100%.

Absolute deviation 5 Raul’s score 2 average score 5 90 2 85 5 5.

Percent deviation 5 5 ___ 85 × 100% 5 500% ÷ 85 5 5.88% (approximately).

5.88% is closer to 5.9% than to 5.8%, so 5.9% is correct. (117)

43. Choice B is correct. If we discount 20% and then 12%, we are, in effect, taking 88% of 80% of the origi-nal price. Since “of ” represents multiplication, when we deal with percent we can multiply 88% 3 80% 5 70.4%. This is a deduction of 29.6% from the original price. (119, 114)

44. Choice A is correct.

This is a simple proportion: 1 foot ______ 1 __ 2 mile

5 1 __ 2 inch

______ x . Our

first step must be to convert all these measure-ments to one unit. The most logical unit is the one

our answer will take—feet. Thus, 1 foot _________ 2,640 feet 5

1 ___ 24

foot _______ x . (1 mile equals 5,280 feet.) Solving for x, we

find x 5 2,640 _____ 24

feet 5 110 feet. (120, 121)

45. Choice A is correct. Let the two original sides of the rectangle be a and b and the new sides be a' and b'. Let side a increase by 25%. Then

a' 5 (100 1 25)% a 5 125% a 5 125 ____ 100 a 5 1.25a

5 5a ___ 4 . We also have that ab 5 a'b'. Substituting

a' 5 5a ___ 4 , we get ab 5 5a ___ 4 b'. The a’s cancel and we get

b 5 5 __ 4 b'. So b' 5 4 __ 5 b, a decrease of 1 __ 5 , or 20%. (122)

46. Choice C is correct. Using a calculator, we get: 6,043 _____ 2,071

5 2.9179 for Choice A; 4,290 _____ 1,463 5 2.9323 for Choice

B; 5,107 _____ 1,772 5 2.8820 for Choice C; 8,935 _____ 2,963 5 3.0155 for

Choice D; and 8,016 _____ 2,631 5 3.0467 for Choice E. Choice

C has the smallest value. (104, 127).

47. Choice D is correct. Let’s say that the price was $100 during 2003. 30% of $100 5 $30, so the new price in 2003 was $130. In 2004, the company cut back its prices 20%, so the new price in 2004 5

$130 2 ( 20 ____ 100 ) $130 5

$130 2 ( 1 __ 5 ) $130 5

$130 2 $26 5 $104.The net change is $104 2 $100 5 $4.

$4 _____ $100

5 4% increase (118)

48. Choice E is correct. 0.04% 5 0.04 ____ 100 5 4 ______ 10,000

5

1 _____ 2,500 . (107)

49. Choice D is correct. Before adding you should examine the numbers to be added. They form pairs, like this: 16 1 (12 1 88) 1 (34 1 66) 1 (21 1 79) 1 (11 1 89), which equals 16 1 100 1 100 1 100

1 100 5 416. Dividing 416 by 25, we obtain 16 16 ___ 25 ,

which equals 16.64. (112)

50. Choice A is correct. We can set up a proportion as follows:

1 coconut _________ 1 banana

5 2 __ 1 , 1 banana ___________

1 grapefruit 5 1 __ 3 , so by multiplying

the two equations together

( 1 coconut _________ 1 banana

× 1 banana ___________ 1 grapefruit

5 2 __ 1 × 1 __

3 ) and cancel-

ing the bananas and the 1’s in the numerators and

denominators, we get: 1 coconut ___________ 1 grapefruit 5 2 __

3 , which can

be written as 2 : 3. (120)

SAT2015_P06.indd 194 4/23/14 11:39 AM

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