problms involved with real numbers

Problems Involved with

Whole Numbers, Decimals, and Fractions

How to add whole numbers?

The first step is to line up the numbers vertically so

that the units digits are in the same column. Next,

add the units digits, the tens digits, and the

hundreds digits.

Adding whole numbers

Each number being added is called an addend and the total, which is the answer to the addition problem is called sum.Example:

6 addend +_4__ addend

10 sum

Problem Solving: Adding whole numbers

1. Jose sold 91 apples, 150 oranges and 141 pears. How many pieces of fruit did he sell altogether?

Solution: 150

+141_ 91382

Answer: Jose sold 382 pieces of fruit altogether.

Another problem…

2. Farmer Ben planted 7 992 seeds in the first half of the year and 1 466 in the second half. How many seeds did he plant in the year?

Solution:7 992

+1 4669 458

Answer: Farmer Ben planted 9 458 seeds in the year.

How to subtract whole numbers?

Write the smaller number under the larger, taking

care to align the same units. Then, starting with the

ones on the right, subtract each digit on the bottom

from the corresponding digit on top. When the

bottom digit is greater, consider the top digit

increased by 10. To compensate, add 1 to the next

bottom digit.

Subtracting whole numbers

Subtracting whole numbers is the inverse operation of adding whole numbers.Instead of adding two numbers to get a sum, you are removing one number from another to get a difference. First, look at the following simple subtraction problem.Example:

8 minuend-4_ subtrahend4 difference

Problem solving: Subtracting whole numbers

1. Jessica has 1 135 beads. 604 beads are red and the rest are blue. How many blue beads does she have?

Solution:1 135

- _604531

Answer: Jessica have 531 blue beads.

Another problem…

Beth and Ken donated Php 2 300 to a charitable organization. Ken donated Php 658. How much did Beth donated?

Solution:2 300

-_ 658_1 642

Answer: Beth donated Php 1 642 to a charitable organization.

How to multiply whole numbers?

Align the multiplier (on the bottom) with the ones digit

of the multiplicand (on top), and draw a line. Then

multiply each digit of the multiplicand. Write the ones

digit of each product below the line.

If there is a tens digit, carry it -- add it -- to the next

product.

Multiplying whole numbers

The basic idea of multiplication is repeated addition.Example:

5 factorsx 3_ factors15 product

Problem Solving: Multiplying whole numbers

1. An apartment has 4 bedrooms. Each bedroom has 3 bookcases. How many bookcases are there in the apartment?

Solution: 4

_x3_12

Answer: There are 12 bookcases in the apartment

Another problem…

2. There are 60 minutes in 1 hour. How many minutes are there in 12 hours?

Solution:60

_x12_720

Answer: There are 720 minutes in 12 hours.

How to divide whole numbers?

The problem of division is to find what number times

the Divisor will equal the Dividend. That number is

called the Quotient. To find the quotient, there is a

method called short division.

Dividing whole numbers

Division is the inverse of multiplication; therefore it depends on knowing the multiplication table.Example:

2 quotient8 16 dividend

-16__0

divisor

Problem Solving: Dividing whole numbers

1. In one night, a movie theater sells tickets for 6 450 dollars. Each ticket costs 15 dollars. How many people purchased a ticket?

Solution: 43015 6 450

-60___45

-45__0

Answer: There are 430 people who purchased a ticket.

Another problem…

2. How many hours are there in 660 minutes?

Solution: 1160 660

-60__60

_-60_0

Answer: There are 11 hours in 660 minutes.

How to add decimals?

To add decimals, follow these steps:

1. Write down the numbers, one under the other, with

the decimal points lined up.

2. Put in zeros so the numbers have the same length

3. Then add using column addition, remembering to

put the decimal point in the answer.

Adding DecimalsDecimal. Based on 10. Example: the numbers we use in everyday life are decimal numbers, because there are 10 of them (0,1,2,3,4,5,6,7,8 and 9). Often "decimal number" is also used to mean a number that uses a decimal point followed by digits that show a value smaller than one. Example:

8.3 addend+4.7_ addend13.0 sum

Problem Solving: Adding Decimals

Problem #1: Ellen wanted to buy the following items: A DVD player for $49.95, a DVD holder for $19.95 and a personal stereo for $21.95. Does Ellen have enough money to buy all three items if she has $90 with her?

Analysis: The phrase enough money tells us that we need to estimate the sum of the three items. We will estimate the sum by rounding each decimal to the nearest one. We must then compare our estimated sum with $90 to see if she has enough money to buy these items.

Answer: No, because rounding each decimal to the nearest one, we get an estimate of $92, and Ellen only has $90 with her.

Another problem…

Problem #2: Melissa purchased $39.46 in groceries at a store. The cashier gave her $1.46 in change from a $50 bill. Melissa gave the cashier an angry look. What did the cashier do wrong?

Analysis: We need to estimate the difference to see if the cashier made a mistake.$50.00 - $40.00 = $10.00

Estimate: $1.46 is much smaller than the estimated difference of $10.00. So the cashier must have given Melissa the wrong change.

How to subtract decimals?

To subtract decimals, follow these steps:

1. Write down the two numbers, one under the other,

with the decimal points lined up.

2. Add zeros so the numbers have the same length.

3. Then subtract normally, remembering to put the

decimal point in the answer.

Subtracting Decimals

Subtracting decimals is easy when you keep your work neat.Example:

67.9 minuend_-23.2_ subtrahend

44.7 difference

Problem Solving: Subtracting decimals

Problem #1: Drake has 2.5 million in his bank account, he withdraw 1.3 million to buy a house and lot for his family. How much money does he have left?

Solution:2.5

-1.3_1.2

Explanation: The total amount of money Drake has in his bank account is 2.5 million, since he withdrawn 1.3 million to buy a house and lot, the remaining amount in the bank is now 1.2 million.

How to multiply decimals?

Just follow these steps:

1.Multiply normally, ignoring the decimal points.

2.Then put the decimal point in the answer - it will have

as many decimal places as the two original numbers

combined.

Multiplying Decimals

When you multiply decimals, you multiply them the exact same way you would multiply whole numbers. Then you count the number of spaces you have in your 2(two) numbers to multiply and you got to have that many spaces in your product.

Example: 32.12 factor

_x0.5_ factor

16.06 product

Problem solving: Multiplying decimalsProblem #1:Two students multiplied 0.2 by 0.4. Student 1 found a product of 0.8 and Student 2 found a product of 0.08. Which student had the correct answer? Explain.

Student 1: 0.8 Student 2: 0.08

Analysis: Let's convert each decimal to a fraction to help us solve this problem.

Fractions: 2/10 x 4/10= 8/100Decimals: 0.2 x 0.4 = 0.08If we multiply two tenths by four tenths, we get a product of eight hundredths.Answer: Student 2 is correct since 0.2 x 0.4 = 0.08.

How to divide decimals?

To divide decimal numbers:

1. If the divisor is not a whole number, move decimal

point to right to make it a whole number and move

decimal point in dividend the same number of places.

2. Divide as usual. ...

3. Put decimal point directly above decimal point in the

dividend.

4. Check your answer.

Dividing Decimals

The picture above shows how to divide decimals.

Problem Solving: Dividing decimals

School lunches cost $14.50 per week. About how much would 15.5 weeks of lunches cost?Analysis: We need to estimate the product of $14.50 and 15.5. To do this, we will round one factor up and one factor down.Estimate:

$14.50 $10x15.5 _20_

Answer: The cost of 15.5 weeks of school lunches would be about $200.

How to add fractions(similar denominators)?

Instructions for adding fractions with the same

denominator

1. Build each fraction (if needed) so that both

denominators are equal.

2. Add the numerators of the fractions.

3. The new denominator will be the denominator of

the built-up fractions.

4. Reduce or simplify your answer, if needed. Factor

the numerator.

How to Add Fractions with Different Denominators?

When the fractions that you want to add have

different denominators, there are a few different ways

you can do it. Here, you’ll learn the easy way, then a

quick trick that works in a few special cases, and

finally, the traditional way.

Here’s some ways to do it:

1. Cross-multiply the two fractions and add the results

together to get the numerator of the answer.

Suppose you want to add the fractions 1/3 and 2/5. To

get the numerator of the answer, cross-multiply. In other

words, multiply the numerator of each fraction by the

denominator of the other.

2. Multiply the two denominators together to get

the denominator of the answer.

To get the denominator, just multiply the

denominators of the two fractions.

3. Write your answer as a fraction.1 + 2 = _1(5) +_2(3)_ = 5 + 6 = 11

3 5 15 15 15 15 15

When you add fractions, you sometimes need to reduce

the answer that you get. Here’s an example:

Because the numerator and the denominator are both

even numbers, you know that the fraction can be

reduced. So try dividing both numbers by 2:

This fraction can’t be reduced further, so 37/40 is

the final answer.

In some cases, you may have to add more than one

fraction. The method is similar, with one small

tweak.

1. Start out by multiplying the numerator of the first

fraction by the denominators of all the other

fractions.

(1 5 7) = 35

2. Do the same with the second fraction and add

this value to the first.

35 + (3 2 7) = 35 + 42

3. Do the same with the remaining fraction(s).

35 + 42 + (4 2 5) = 35 + 42 + 40 = 117

When you’re done, you have the numerator of the

answer.

4. To get the denominator, just multiply all the

denominators together:

Complete Solution: 1 +3+4 = 1(35) + 3(14) + 4(10)

2 5 7 70 70 70

= 35 + 42 + 40

70 70 70

= 117

70

Problem Solving: Adding Fractions

Example #1: John walked 1/2 of a mile yesterday and 3/4 of a mile today. How many miles has John walked?

Solution:

This word problem requires addition of fractions.

Choosing a common denominator of 4, we get.

1/2 + 3/4 = 2/4 + 3/4 = 5/4So, John walked a total of 5/4 miles.

Example #2:Mary is preparing a final exam. She study 3/2 hours on Friday, 6/4 hours on Saturday, and 2/3 hours on Sunday. How many hours she studied over the weekend?

Solution

Choosing a common denominator of 12, we get:

3/2 + 5/4 + 2/3 = 18/12 + 15/12 + 8/12 = 41/12 = 3.42 hours So, Mary studied a total of 3.42 hours.

How to subtract decimals(similar denominators)?

There are 3 simple steps to subtract fractions

1. Make sure the bottom numbers (the denominators)

are the same.

2. Subtract the top numbers (the numerators). Put the

answer over the same denominator.

3. Simplify the fraction (if needed).

Subtracting Fractions with dissimilar denominators.

First of all, when subtracting fractions with different

denominators, the first step in the Rule says that we must

change these fractions so that they have the “same

denominator”.

Here are the steps for subtracting fractions with different

denominators.

So, here are the steps.

1. Build each fraction so that both denominators are

equal. Remember, when subtracting fractions, the

denominators must be equal. So we must complete this

step first. What this really means is that you must find

what is called a Common Denominator. Most of the

time you will be required to work the problem using

what’s called the Least Common Denominator (LCD).

In either case you will build each fraction into an

equivalent fraction.

2. Re-write each equivalent fraction using this new

denominator

3. Now you can subtract the numerators, and keep the

denominator of the equivalent fractions.

4. Re-write your answer as a simplified or reduced

fraction, if needed.

Problem Solving: Subtracting Fractions

Example #1: A recipe needs 3/4 teaspoon black pepper and 1/4 red pepper. How much more black pepper does the recipe need?

This fraction word problem requires subtractionSolution:

The fact that the problem is asking how much more black pepper the recipe needs is an indication that 3/4 is bigger than 1/4However, it does not hurt to check.

3/4 - 1/4 = 2/4 = 1/2The black pepper is 1/2 of a teaspoon more than the red pepper.

Example #2:A football player advances 2/3 of a yard. A second player in the same team advances 5/4 of a yard. How much more yard did the second player advance?

Again, we need to perform subtraction to solve this problem.Solution:

5/4 - 2/3 = 15/12 - 8/12 = 7/126/12 is = 1/2, so 7/12 is just a bit more than half.

So, the second player advanced by about half of a yard more.

Thank you for participating and God bless you all!