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1. RATIOS AND PROPORTIONS

The ratio of two numbers  and  is the fractiona

b . We use ratios to compare

numbers.

A proportion is the equality between two ratios,a c

b d =

 .  and  are called the

extremes and  and  the means.

Example:

2 6

3 9=  is a proportion.

When one of the four numbers in a proportion is unknown you can use cross-multiplication to nd it.

Example: Find :

55 18 3 90 3 30

3 18

 x x x x= ⇒ ⋅ = ⋅ ⇒ = ⇒ =

2. DIRECT PROPORTION

A magnitude is in direct proportion to another one:■ when you multiply one quantity by a number (double, triple, etc), you

multiply the other quantity by the same number (double, triple, etc).■

 when you divide one quantity by a number (half, third, etc.), you dividethe other quantity by the same number (half, third, etc).

Examples:a) The number of chocolate bars you buy is in direct proportion to the price.

(1 chocolate bar costs 1.50 €; 2 chocolate bars cost 3 € ; ... ).b) The number of hours you work is in direct proportion to your salary.

Exercise: Complete this proportionality table:The price of a cake is 1.50 €.

Cakes 1 3 7Price 1.50 7.50 15

 Keywords 

Ratic   Proportion Mean  Extreme 

Rule of thRee  

Constant of proportionality 

 Per cent  

 Percentage  oUt of 

Increase 

 Decrease

LESSON 4: PROPORTIONALITY

 AND PERCENTAGES 

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Notice that if you divide the price by the number of cakes you always get thesame result, it is called the constant of proportionality.

To solve problems about direct proportion we use the direct rule of three.

Example:4 books cost £60. How much are 15 books?

4 £60 4 60 60 15( . .) 4 60 15

15 15 4

books D P x x

books x   x

→     ⋅⇒ = ⇒ ⋅ = ⋅ ⇒ = =

→    

£225

3. INVERSE PROPORTION

Two numbers are in inverse proportion if a quantity increases at the samerate as the other quantity decreases (if you multiply a quantity by two, thenthe other quantity is divided by two).

Examples:

a) The number of workers is in inverse proportion to the number of days theyneed to do a job. (The more workers there are, the fewer days they need).

b) The speed of a car is in inverse proportion to the time to reach a certaindistance.

Exercise: Complete this proportionality table:

Workers 1 2 4

Hours 12 4

Notice that if you multiply the number of workers by the number of hours youalways get the same result.

To solve problems about direct proportion we use the inverse rule of three.

Example:4 men do a job in 12 days. How many days do 6 men need to do the same job?

4 12   4 4 12( . .) 4 12 6

6   6 12 6

men days   x I P x x

men x

→     ⋅⇒ = ⇒ ⋅ = ⋅ ⇒ = =

→    

8 days 

4. MORE THAN ONE PROPORTIONAL RELATIONSHIP

Sometimes, there is more than one proportional relationship, then you haveto check if the relationships are direct or inverse, then you solve the problemas shown:

Example:A farmer needed 294 kg of grain to feed 15 cows for a week. How manykilograms of grain does he need to feed 10 cows for 30 days?

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Three magnitudes: kilograms of grain, cows and days.Direct

COWS DAYS KG

15  ─────  7  ─────  294

10  ─────  30  ─────   

Direct

You have to multiply both proportions, so

294 15 7 294 105 294 300840

10 30 300 105 x kg 

 x x

⋅= ⋅ ⇒ = ⇒ = =

Notice: If there is an inverse proportion, you have to use the reciprocal fractionfor that magnitude.

5. PERCENTAGES

Per cent means "out of 100". We use the per cent symbol (%) to representpercentages. You can say "8 out of 100” or "8 per cent" (8%).

There are three ways to write the same thing: percentages, fractions withdenominator 100 and decimal numbers:Example:

1515% 0.15

100= =

You can write a decimal number as a percentage (by moving the decimal point

two places to the right) and vice-versa.

Example: 0.85 = 85%

5.1. Calculating percentages of numbers

You can nd any percentage of a number by converting the percentage into adecimal number and multiplying.

Example: Let's calculate 30 percent of 400:

Option 1: Using percentages and fractions

30% of 400 =30

100 · 400=

30 400

100

⋅ =120

Option 2: Using decimal numbersFirst change 30% to a decimal number by moving the decimal point 2 placesto the left

30% = 0.30

Then multiply: 0.30 · 400 = 120So, we have: 30% of 400 is equal to 120.

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5.2. Easy percentages

■ 50% =1

2 (dividing by 2)

  ■ 25%=1

4 (dividing by 4)

  ■ 20%=1

5 (dividing by 5)

5.3. Percentage increase and percentage decrease

To calculate percentage increase and percentage decrease you must rememberthis:

 add for an increase

  andsubtract for a decrease

Example: (PERCENTAGE INCREASE)

An antique jug costs 25% more in 2009 than in 2007. The original price was£40. How much does it cost now?

First, you nd the increase: 25% of £40 = 0.25 · £40= £10Then, you add this to the original price: £40 + £10 = £50The jug costs now £50

Example: (PERCENTAGE DECREASE)

This mobile phone costs £60, but it is reduced by 25%. Howmuch is it now?

First, you nd the decrease: 25% of £60 =0.25 60 = £ 15Then, you subtract this to the original price: £60 - £15 = £45So the sale price is £45

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Exercise 1. Calculate the unknown number in these proportions

a)3 6

2   x=   b)

2

1.5 2

 x=  

c)21

9 15

 x=   d)

3 2

3 x

=  

Exercise 2. Which number has the same ratio to 15 as 5 to 10?

Exercise 3. An athlete runs 42 kilometres in two and a half hours. How much will herun in 10 hours at the same speed?

Exercise 4. There is a fountain ten metres away from an empty olympic-size swim-ming-pool. We have a one-litre jug to ll the swimming-pool. The pool is 50 m long,25 m wide and 2 m deep. It takes 2 minutes to ll the jug from the fountain, pourthe water from the jug in the pool and go back. How long does it take to ll up theswimming-pool?

Exercise 5. We have paid 15.80 € for 315 photocopies in a copy centre. How muchwill we pay for 10 photocopies?

WORKSHEET 

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Exercise 6. For cooking meatballs for 6 people the recipe says that we need 500 gof minced beef. How much minced beef do we need if we want to cook meatballs for4 people?

Exercise 7. A toner cartridge of a laser printer costs 60 € and can print 3500 copies.How much does each copy cost?

Exercise 8. My car uses 9 litres of petrol to travel 100 km.a) How far can I travel with 63 litres?b) How much petrol would I need to travel 220 km?

Exercise 9. It takes Larry 1h 30 min to drive from Albacete to Valencia at an aver-

age speed of 120 km/h. How long does the same journey take at an average speedof 110km/h?

Exercise 10. A farmer has 640 sheep and he can feed them for 60 days. How many

sheep must he sell if he wants to feed them for 75 days with the same amount offood?

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Exercise 11. In Massanassa (a village in Valencia) the population is 1200 peopleand 300 tourists went there to celebrate a party. In the party, they all started to eata 50-kg paella at 10:00. They nished the paella at 15:00. How long do 500 peopleneed to eat a paella of 70 kg?

Exercise 12. A naughty student has to copy this line one hundred times: “I will notmake fun of my Maths teacher.” It takes him 60 minutes.

a) How long does it take for three students to copy this sentence ve hundred

times?

b) How long does it take for the whole class (of 30 pupils) to copy “I will neverforget to do my Maths homework” one thousand times. (Note: It is supposedthat the number of letters of each sentence is in direct proportion to the timeit takes to copy it).

Exercise 13. In a TV contest you have the chance of choosing a prize:

Prize 1: 25% of 500€; Prize 2: 30% of 400€; Prize 3: 90% of 150€

Which one do you prefer?

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Exercise 14. In a sale the price of a pair of shoes is 50€ which is 75% of the usualprice. What was the original price?

Exercise 15. Thirty-two per cent of the students in the class have failed the lastMaths test and there are exactly 12 pupils who must retake the test, what is the totalnumber of students in the class?

Exercise 16. The price of a wardrobe before taxes is 480 € plus 18% VAT and thesalesman offers a 12% discount. What is the nal price?

Exercise 17. I have paid 60 € for a pair of glasses and the original price was 75 €.What is the percentage of discount?