Proportions and percentages

Index Ratio Proportion Percentages Exercises

Proportions and percentages

Matematicas 2o E.S.O.Alberto Pardo Milanes

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1 Ratio

2 Proportion

3 Percentages

4 Exercises

Alberto Pardo Milanes Proportions and percentages


Ratio



Ratio

What’s an Ratio?

A Ratio is a comparison of two or more quantities. Ratios can beused to compare costs, weights, sizes and other quantities.Each number in a ratio is called a term. You can write ratios indifferent ways:

Example:12 parts to 25 parts,12 to 25,12 : 25,12

25,. . .



Ratio

Equivalent Ratios

Equivalent Ratios (or Equal Ratios) are ratios that mean the same.You can simplify a ratio to its lowest terms, by finding the so calledsimplest form of a ratio. A ratio is normally written using wholenumbers only(in its simplest form).Ratios can also be written in the form 1 : n or n : 1 (n is writtenas a decimal). When one of the numbers in a ratio is 1, the ratio isin Unitary form.

Examples: Twenty to forty, is equivalent to five to ten. One totwo is the simplest form for twenty to forty. 0.5 to 1 is the unitaryform for twenty to forty.



Proportion



Proportion

A proportion is an equation stating that two ratios are equal. Equalratios have equal cross products; equal cross products show that aproportion is true.

Examples:

• 14

21=

10

15because 14 · 15 = 21 · 10 = 210.

• 1 · 6 = 2 · 3 because1

2=

3

6.



Proportion

Two quantities are in direct proportion if their ratio stays the sameas the quantities increase or decrease.

Examples:• You drive a car at 70 miles per hour. The miles are directlyproportional to the hours you drive, so in two hours you will drive140 miles, in six hours you will drive 420 miles.• Three pens cost nineteen euros and fifty cents. The cost is direct-ly proportional to the number of pens, so two pens cost thirteeneuros.

Two quantities are in inverse proportion when one increases at thesame rate as the other decreases.

Example: A farmer has enough food to feed 200 chickens for 20days. If he buys 200 more chickens, how long would the sameamount of food feed the chickens?



Percentages



Percentages

A percentage ( %) is a fraction out of 100. To find a percentage ofa quantity change the percentage to a fraction or a decimal andmultiply it by the quantity.

Example: 75% means75

100=

3

4= 0.75. So 75% of 200 euros

=75

100· 200 =

15000

100= 150 euros.

To increase a quantity by a percentage you have to find thepercentage of the quantity and add it to the original quantity. Todecrease a quantity by a percentage find the percentage of thequantity and subtract it from the original quantity.



Percentages

Examples:

• If you add a 12% tip to a 20 bill, find 12% of 20 =12

100· 20 =

0.12 · 20 = 2.4 and add it to the bill: 20 + 2.4 = 22.4 euros. Youcould also find 112% of 20 = 22,4 euros, because 100+12 = 112.• A pair of shoes costs 25 euros. It is on a sale for 45% off. To find

the sale price, find 45% of 25 =45

100· 25 = 0.45 · 25 = 11.25 and

take it away from the original price: 25−11.25 = 13.75 euros. Youcould also find 55% of 25 = 13.75 euros, because 100−45 = 55.



Exercises



Exercises

Exercise 1

Are those quantities in direct proportion?

• Fifteen boys to one hundred girls and twenty-five boys to fortygirls:

• Four dogs to six cats and forty-two dogs to twenty-eight cats:

• Five cars to four trucks and fifty-five cars to forty-four trucks:



Exercises

Exercise 2

Team A won 12 games and lost 18 games, while team B won 12games and lost 10 games. Write those ratios in unitary form.Which team had a higher ratio of wins to losses?



Exercises

Exercise 3

Six books cost 42 euros. How much will ten books cost?



Exercises

Exercise 4

My pie’s recipe calls for 2 kilograms of apples and 3 kilograms ofpeaches. My sister’s recipe requires 1 kilogram of apples and 2kilogram of peaches. Which recipe has a higher ratio of apples topeaches?



Exercises

Exercise 5

Lou spent 4 minutes making 2 phone calls. Assuming therelationship is directly proportional and that he makes 24 phonecalls;how much time will he spend in total?



Exercises

Exercise 6

A field provides enough food for 25 donkeys for eight days. Forhow long would the same field feed 20 donkeys?



Exercises

Exercise 7

A CD normally cost 22 euros. In a sale there is a reduction of 25%.What is the sale price? The price of a DVD has been reduced by25% to 15 euros in a sale. What was the original price?



Exercises

Exercise 8

A factory employs 200 workers. Next year the numbers of workerswill increase by 20% How many workers will there be?



Exercises

Exercise 9

A bicycle costs 132 euros. It can also be bought with a deposit of15%, followed by six monthly payments of 21,50 euros. Find thetotal cost of buying on credit and the difference between the cashprice and the cost of buying on credit.


Education

Proportions and percentages