Maths Module 3 - James Cook University

Maths Module 3

Ratio, Proportion and Percent

This module covers concepts such as: ? ratio ? direct and indirect proportion ? rates ? percent

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Module 3

Ratio, Proportion and Percent

1. Ratio 2. Direct and Indirect Proportion 3. Rate 4. Nursing Examples 5. Percent 6. Combine Concepts: A Word Problem 7. Answers 8. Helpful Websites

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1. Ratio

Understanding ratio is very closely related to fractions. With ratio comparisons are made between

equal sized parts/units. Ratios use the symbol " : " to separate the quantities being compared.

For example, 1:3 means 1 unit to 3 units, and the units are the same size. Whilst ratio can be

expressed as

a fraction,

the ratio 1:3

is NOT the

same as

1 3

,

as

the

rectangle

below illustrates

The rectangle is 1 part grey to 3 parts white. Ratio is 1:3 (4 parts/units)

The rectangle is a total of 4 parts, and therefore, the grey part

represented

symbolically

is

1 4

and

not

1 3

AN EXAMPLE TO BEGIN:

My two-stroke mower requires petrol and oil mixed to a ratio of 1:25. This means that I add one part

oil to 25 parts petrol. No matter what measuring device I use, the ratio must stay the same. So if I

add 200mL of oil to my tin, I add 200mL x 25 = 5000mL of petrol. Note that the total volume (oil and

petrol combined) = 5200mL which can be converted to 5.2 litres. Ratio relationships are

multiplicative.

Mathematically:

1:25 is the same as

2:50 is the same as

100:2500 and so on.

To verbalise we say 1 is to 25, as 2 is to 50, as 100 is to 2500, and so on......

Ratios in the real world:

House plans 1cm:1m = 1:100 Map scales 1:200 000 Circles C:D is as : 1 Golden ratio : 1 as is 1.618:1often used in art and design as

shown right with the golden rectangle. This is also the rectangle ratio used for credit cards.

Key ideas:

Part-to-part ratio relationships Comparison of two quantities (e.g. number of boys to girls) A ratio is a way of comparing amounts A ratio shows the number of times an amount is contained in another, or how much bigger one amount is than another The two numbers are both parts of the whole If I need to mix some cement, then I could add two parts cement to four parts sand. Hence the ratio 2:4 (6 parts in total). The written expression is important; 4:2 would give a different mix.

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 Ratios are written to their simplest form. In the figure to the right we have 15 red dots and five green dots. The ratio is 15:5; however, this can be reduced to the simplest form as we do with fractions. The ratio as we can see in the graphic is also 3:1, if you look at the relationship of the numbers 15:5 as is 3:1, we can see that 3 is multiplied by 5 to get 15 and so is the one to get five.

We can also think of the ratio as being a part of the whole 100%, so in the instance above, we would have 75%:25% (each being a part of 100%). If we had 100 dots, 75 of them would be red and 25 of them would be green.

Part-to-whole ratio relationship: (e.g. boys to class)

15:5 is as 3:1

One quantity is part of another; it is a fraction of the whole. For example, there are 12 boys as part of 30, which is written 12:30

1. Your Turn:

Write down as many things as you can about the ratio of the table above. a. What is the part to part ratio? b. What is the part to whole ratio? c. What is the ratio of part to part in the simplest form? d. What is the ratio of part to whole in the simplest form? e. What is the ratio as a percentage for part to part?

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2. Direct and Indirect Proportion

Indirect proportion ? sometimes called `inverse proportion'

When a variable is multiplied by a number and the other variable Num. of

is divided by the same number ? they are said to be in indirect

people

8

4 2 1

proportion. For example, it takes 8 people ? hour to mow the

lawn, 4 people take one hour, 2 people 2 hours, and 1 person 4 Time in 1/2 1 2 4

hours.

hours

Direct proportion

When one variable changes, the other changes in a related way; the change is constant and multiplicative. For example, if we look at the ratios, 1:25 and 2:50 we multiply both variables by two.

1: 25, 1 ? 2 = 2 25 ? 2 = 50 2: 50.

Let's look at this again in a practical situation: A man plants two trees every ten minutes, four trees every 20 mins, six trees every 30 mins... 2:10, 4:20, ...

We can measure the height of a tree on a sunny day without having to go up a ladder. We simply need to measure the height of a stick, in the same vicinity, and the shadows that they both cast. Here we are dealing with the cross products (meaning the multiplicative relationship) of 2 equivalent ratios.

For example, if a tree shadow is 13m, and we measure a shadow of a 30cm upright ruler in the same vicinity we can work out how tall the tree is without climbing a ladder! But first we will convert to the same unit. The stick is 30cm and its shadow is 50cm, thus 13m for the shadow of the tree is converted to 1300cm.

30cm 50cm

13m (1300cm)

If we know that 30 50 is as 1300, then we can write

30 50

1300

Next we can use a method of cross multiplication because ? 50 30 ? 1300

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