Unit 5:N u m e r i c a lProportionality

In this lesson you will learn about: Proportions and their properties Directly and inversely proportional

magnitudes Percentages

Vocabulary you need in this lesson:You are going to use some vocabulary related to this lessonand the problems you will work with. The same words appe-ar in the crossword, in the word scramble and in the wordsearch.

Some theoryA proportion is an equation that states that two ratios are

equal, such as

.

Property :in a true proportion, the product of themeans equals the product of the extremes

If two quantities are in DIRECT proportion, when onequantity is doubled, so is the other, i.e. they both in-crease in the same proportion. If two quantities are inDIRECT proportion, their quotient is a constant, i.e. ifyou divide one by the other you always get the sameresult.

If two quantities are in INVERSE proportion to eachother, then if one gets doubled, the other gets halved.If two quantities are in INVERSE proportion, theirproduct is a constant, i.e. if you multiply them togetheryou always get the same result.

A percentage is a way of expressing a number as afraction of 100 (per cent meaning “per hundred”). It isoften denoted using the percent sign “%”. For example,45% (read as “forty-five percent” is equal to 45/100,or 0.45)

Now it´s time to practise

1. Exercises

1) In the Formula-One Grand Prix from Australia, 22 cars were competing. Four ofthem withdrew from the race. What is the ratio of the cars which finished therace?

2) Find the proportions which are OK among the following:

b) c) d)

3) Complete the following equalities so the corresponding ratios are proportional:

4) Find the proportions which are true among the following

5) Complete the following series:

6) Find the value of x in the following

2. Problems

1) Two burglars are sharing a haul using the ratio 7:11.

The first of them receives €84000. How much money does the second receive?

2) Indicate if the following quantities are direct or inverse proportions:

- Number of sheets you use and the number of boogers you produce during a cold.

- Meters of toilet paper you use and the number of times you go to the toilet during adiarrhea attack. (Suppose you use the same quantity of paper every time you go tothe toilet).

- The number of people which are eating a leg of ham and the weight of ham each oneeats.

- The cholesterol level you have and the number of Bollycaos you eat a day.

- The cholesterol level you have and the number of hours of exercise you do during aday.

- The number of fish you catch and the number of fishing hooks you have on the line.

- The money you have and how far can you go on holiday.

- The amount of money each worker earns and the number of workers a company cansupport.

3) In the famous restaurant Doña Paca’s House, they use up four legs of ham every ninedays. How many legs of ham will they use during February and March? Sol:26.22

4) A flying donkey whose speed is 90 km/h , runs 160 km. How far would the donkey run ifits speed were 50 km/h?Sol:88.88 km

5) Five bricklayers build a wall which is 240 meters long, in 6 days. How many days willtwelve bricklayers spend building the same wall? Sol: 2.5 days

6) Count Dracula has set up a nursery to look after little vampire children from Transylva-nia. There are 150 little vampires at the nursery. Dracula uses a tank of blood tofeed them during 27 days. Due to the current crisis, 30 new vampire children havecome to Dracula’s Nursery School because their parents have to work collecting rad-ishes at Transylvanian Countryside (“Vampires can’t only live on blood”, vampires usedto say).

How many days will Count Dracula be able to feed his charming little bloodsuckers withthe same tank of blood? Sol: 22.5 days

.

7) Jacinto Pelopijo has many lice on his hair. Manuelo Sorullez is a louse who works in areal state company, which is dedicated to buying and selling plots of hair .

Every louse who wants to build a house on somebody’s head can talk to Mr. Sorullez inorder to buy and finance a beautiful and pleasant plot of hair.

Mr. Sorullez has sold 5200 plots of hair on Pelopijo’s head, so , each family of lice hasan area of hair of 0.064 cm2 (all of Pelopijo’s head’s surface has already been sold).

Bartolomero Calvador shaved his head recently, so ,1500 families of lice have to moveto a new head. All of them moved to Pelopijo’s head, so they have to share Pelopijo’shead with lice that were there already.

How big is each plot of hair now? Sol: 0.0497 cm2

8) Darth Báter has been fighting against The Alliance Rebellion’s troops 8 hours a day for16 days. He destroyed 1368 Rebel’s starships during this time.

How many Rebel starships would Darth Báter have destroyed if he had been fighting12 hours a day over 20 days?

9) Three troglodytes have caught a mammoth whose weight is 5400 kg. They have decidedto divide it up according to their weights, which are the following:

Peter Flintstone…………………….. 89 kg

Paul Stonelover……………………… 77 kg

Inodor Fartlauncher………………. 93 kg

How much mammoth did each troglodyte receive in kgs? Sol:1855.6;1605.4;1939.

10) Mr. Marcelino Morcillónez, who actually is “Marquis Of Green Bud”, has four childrenwho aren’t very good students. (In fact they are completely disastrous students):

Joselín ,who has failed 15 subjects,

Rufino ,who has failed 16 subjects,

Jerónima Del Rocío, who has failed 28 subjects,

and Deborah Macarena, who has failed 35 subjects.

Marcelino has a certain amount of money that he has inherited from his father€340000, and he has decided to give each child an amount inversely proportional tothe number of failed subjects that each one of his children has.How much money does each one receive?Sol:117169.23;109846.15;62769.23;50215.39

11) Last year I went to buy some fish for my aquarium. I bought a Red Royal Discus , whoseprice was $129.99, but they charged me only $95.90.

How much was the discount ? Sol: 26.2 %

12) Natives from Amazonia usually sell live fish to be taken to aquariums in North Americaand Europe. Before you buy fish in a pet shop, many people earn money from them.

Before you buy fish , their prices are increased by 45% by the person who takes themfrom the Natives. Next, in Manaos, exporters of live fish increase fish’s prices by 30%.

When live fish arrive in Europe, wholesalers of pets increase their prices by 15%, andfinally, the pet shop’s owner increases fish’s price by 55%.

Suppose that a Heckel Discus is sold by a Native for $3. How much does it cost inHobby Zoo in Carrefour from Camas? Sol: 10.08

14) There is a Detective Agency in Espartinas, which is completely successful, called“Supergabarrinni Snoopers”.They solve every problem that you could have.

Last month, Manolo Escobar went to “SS”(Initials of Supergabarrinni Snoopers), be-cause he wanted to find the nails he had lost many years ago (The famous Manolo’scar’s nails).

Mr Gabarro ‘s taxes are $125 for each nail they find, but they have a special offer :

If they find 2 nails, the total price is 7% less.

If they find 3 nails , the total price is 9% less.

They have to charge VAT( which is 16%).Find the price Manolo has to pay in case SS have found 2 or 3 nails (don’t forget toinclude VAT)Sol: 2 nails….€269.7 3 nails…………€395.85

15) Embutidos Falopalo is a company that makes processed cold meats from Iberian pigs.

After salamis are packed and dried, their weights decrease from drying by 25%.

Before selling salamis, their prices have to be increased by VAT. This is 4%.

My friend Federico bought a piece of salami whose weight was 1.550 kg. What was itsweight before drying?

Prices are fixed usually before drying a product. In other words, they charge us theprice of the salami without dry. If the price of salami is $14.50 per kg, how much will Ihave to pay for a piece whose dried weight is 2.560kg?

Sol:2.067 kg €57.41

16) Aliens have kidnapped 8% of the people who lived in Espartinas.

Today there are 1426 inhabitants in this town. How many inhabitants were there inEspartinas, before they were kidnapped? Sol:1550

Some interesting websites!

If you want to read more about ratios you can go to:

http://www.mathsisfun.com/ratio2.html

To check if you have learnt some concepts correctly go to thefollowing website and do the self-check quizzes:http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-07-829633-1&chapter=7&lesson=3&headerFile=4&state=na

To read a bit more and do some more practice take a look at thispage:

http://www.emathematics.net/porcentajes1.php?p=&d=&tp=1

To practice a little with direct proportion go to:

http://www.emathematics.net/porcentajes1.php?tp=2

To practice a little with inverse proportion go to:

http://www.emathematics.net/porcentajes1.php?tp=3

If you want to read about a very interesting proportion , the golden pro-portion , that you can find in many situations in your daily life, click here:

http://en.wikipedia.org/wiki/Golden_proportion