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Classwork Series and Exercises {Mathematics- SS2}: Ratio, Proportion and Rates

Definition: A ratio is a comparison of two similar quantities obtained by dividing one quantity by the other.

Ratios are written with the : symbol.

Example:

The ratio of 6 to 3 is 6 ÷ 3 = 6/3 = 6 : 3 = 2

Example:

The ratio of 3 to 6 is 3 ÷ 6 = 3/6 = 3 : 6 = 1/2

Since a ratio is only a comparison or relation between quantities, it is an abstract number. For instance, the ratio of 6 miles to 3 miles is only 2, not 2 miles.

As you can see above, ratios can be written as fractions. They also have all the properties of fractions that you have learned in the previous part of this station.

The ratio of 6 to 3 should be stated as 2 to 1, but common usage has shortened the expression of ratios to be called simply 2.

If two quantities cannot be expressed in terms of the same unit, there cannot be a ratio between them.

Can you find these ratios?

1. 5 dollars to 25 cents

5 dollars = 500 cents

so, the ratio of 500 cents to 25 cents = 500 / 25 = 20

2. 4 meters to 3 kilograms

Meters cannot be expressed in terms of kilograms, so this ratio cannot be expressed.

Problem:

If two full time employees accomplish 20 tasks in a week, how many such tasks will 5 employees accomplish in a week?

2 : 5 = 20 : x

2 × x = 5 × 20

This answer is obtained by knowing about proportions and how they are used. You can set up proportions by using ratios. Remember, ratios are comparing similar things. In the problem above, the first ratio is comparing employees and the second is comparing tasks.

Example

If I interviewed 10 dentists and 8 of them approved of a certain breath mint, I could write the ratio 8/10.

Often it does not make sense to reduce a ratio. If I reduced the above ratio I would get 4/5. As a fraction (or as a decimal) this has the same value. But it no longer represents something from the real world. I did not interview 5 dentists, of whom 4 approved!

Similarly I would lie if I changed the ratio to 80/100 this makes it sound like I did a lot more work by interviewing 100 dentists. (Note that many people, especially in advertising, do change ratios in this manner.)

Definition: A proportion is a statement of the equality of two ratios.

Example:

6 : 3 = 2 : 1 or 6 / 3 = 2 / 1or 6/3 = 2/1 are ways to write the proportion expressed as:

6 is to 3 as 2 is to 1

Example:

2 : 8 = 1 : 4 or 2 / 8 = 1 / 4

are ways to write the proportion expressed as:

2 is to 8 as 1 is to 4

If any three terms in a proportion are given, the fourth may be found. Given the proportion:

a : b = c : d  or a / b = c / d

and using the principals of manipulating equations , we observe that

a = (b × c) / d and c = (a × d) / b

b = (a × d) / c and d = (b × c) / a

An easy way to remember this is to say that in a proportion:

The product of the means is equal to the product of the extremes.

It is important to remember that to use the proportion; the ratios must be equal to each other and must remain constant.

EXERCISES

Lets see how much you’ve learnt, attach the following answers to the comment below

1. Find the value of x in 25 : 15 = 10 : x A. 4 B. 5 C. 6  D. 7
2. A pipe transfers 236 gallons of fuel to the tank of a ship in 2 hours. How long will it take to fill the tank of the ship that holds 4543 gallons? A. 45 hrs B. 38.5 hrs C.  57.46 hrs  D.  36 87 hrs
3. A twelve pound shankless ham contains sixteen servings. What is the rate in servings per pound? A. 4/3 B. 3/5 C. 5/4 D. 1/3
4. Use cross products to solve for X, and round to hundredths. 14/38 = X/29 A. 10.684 B. 10.700 C. 10.600  D. 11.500
5. 1 meter equals 100 centimeters. How many meters equal 750 centimeters? A. 8.5 m B. 7.8 m C. 7.5 m  D. 4.5 m