Unitary ratio, direct and inverse proportions

Unitary Ratio, Direct and Inverse Proportions

Prepared by : Mohd Aidil b. OthmanMohd Nur b.

Mohammad Mohd Faiz Ab Rahim

Unitary RatioA ratio in which one of the terms is equal to 1

is called a unitary ratio.

When the first part of ratio is 1, the ratio often referred as unitary ratio.

Example 1Express the following as unitary ratios:a.) 2 : 6 = 2 ÷ 2 : 6 ÷ 2

= 1 : 3

b.) 4 : 9 = 4 ÷ 4 : 9 ÷ 4 = 1 : 2.25

Example 2The ratio of adults to children on a bus is 1 : 5. There are

3 adults, how many children are there?

adult : children = 1 : 51 x 3 : 5 x 3 (to make number of adult 3 )3 : 15

So the number of children is 15.

Example 3The number of boys to girls in a class is 4:9. there are 12

boys, so how many girls are there?

boys : girls = 4:94 ÷ 4 : 9 ÷ 4 (to get unitary ratio of 1 boy)= 1 : 2.251 x 12 : 2.25 x 12 (to make number of boys 12)12 : 27

So the number of girls are 27.

Direct Proportion

When a quantity gets larger or smaller, we say that it changes.

Sometimes a change in one quantity causes a change, or is linked to a change, in another quantity.

If these changes are related through equal factors, then the quantities are said to be in direct proportion. Or one might say that the two quantities are directly proportional.

Example For example, suppose that you are buying cans of soup at the store.

Let us imagine that they cost 50 cents, or RM0.50, each.

Case #1: Suppose that you buy 4 cans. You would pay RM2.00.

Case #2: Suppose that you buy 8 cans. You would pay RM4.00.

So, changing the number of cans that you buy will change the amount of money that you pay.

Notice that the number of cans changed by a factor of 2, since 4 cans times 2 is 8 cans.

Also, notice that the amount of money that you must pay also changed by a factor of 2, since $2.00 times 2 is $4.00. Both the number of cans and the cost changed by the same factor, 2.

The formal definition of direct proportion:

Two quantities, A and B, are in direct proportion if by whatever factor A changes, B changes by the same factor.

Shorthand'sHere is a shorthand way to say that the quantities A and B are

directly proportional:

The Greek letter between A and B is call Alpha. It is here written in lower case script. In this context it is shorthand for

the phrase "is directly proportional to." So, the above statement reads "A is directly proportional to B."

Whenever you have a direct proportion as stated above you can change it into an equation by using a proportionality constant. Here is how the direct proportion would look as an equation:

The above would read "A equals k times B." The quantity k is a proportionality constant.

If two quantities, A and B, are directly proportional, then there is a proportionality constant, k, such that k times B will equal A.

Graph on Direct Proportion If A=kB, then a graph of A vs B will yield a straight line through the

origin with k as the slope:

So, if you graph data for two related quantities, and that graph yields a straight line through the origin, then you know that the two quantities are in direct proportion.

Inverse ProportionProbably better stated as a reciprocal proportion,

the inverse proportions relates two quantities through factors that are multiplicative inverses. That is, through factors that are reciprocals, such as 3 and 1/3.

ExampleFor example, let us say that you are driving a car and you are going

to travel 60 miles. Consider this to be a constant distance throughout the following discussion.

Case #1:

Suppose that you spent 1 hour driving. Your average speed would be 60 mph.

Case #2:

Suppose that you spent 2 hours driving. Your average speed would be 30 mph.

So, changing the number of hours that you drive will change the average speed that you will travel.

Notice that the number of hours, the time, that is, changed by a factor of 2, since 1 hour times 2 is 2 hours.

Also, notice that the speed at which you were traveling changed by a factor of 1/2, since 60 mph times 1/2 is 30 mph.

The two quantities, time and speed, changed by reciprocal factors. Time changed by a factor of 2; speed changed by a factor of 1/2.

When quantities are related this way we say that they are in inverse proportion. That is, when two quantities change by reciprocal factors, they are inversely proportional.

In the above example the time is in inverse proportion to the average speed. One could also say that the average speed was in inverse proportion to the time.

The formal definition of inverse proportion:

Two quantities, A and B, are in inverse proportion if by whatever factor A changes, B changes by the multiplicative inverse, or

reciprocal, of that factor.

Example The rotational speeds of two connected pulleys are inversely proportional to

their diameters. A pulley with a diameter of 4 meter and a speed of 1880 rpm is connected to a pulley with a 6 meter diameter. What is the rotational speed of the second pulley?

Solution :

The pulley diameter increase in the ratio 6 : 4 , therefore the speeds will be related as the inverse ratio 4 : 6 or 2 : 3. Let the speed of second pulley be S. Then expressing the relationship as a proportion, we get

S : 1800 = 4 : 6

7200 = 6S

S = 1200

Thus the speed of the second pulley is 1200 rpm.

The Application of ratios Regular saving Simple and compound interest

Regular SavingsExample A sum of RM 120 is divided between Ali and basri in the ratio 5 : 3. How much will basri receive?

solution

Ali’s share : Basri’s share = 5 : 3

Total number of parts = 5+3

= 8 equal parts

Basri’s share = 3 share

= 3 x RM 15

= RM 45

Therefore Basri will receive RM 45

Simple and compound interesta) Simple interest is the amount paid by bank to the depositor on the principal alone

for a stated period of time

b) Interest rate is the percentage of interest given on principal for a period time

Simple interest = interest rate x principal x time

Interest rate = simple interest x 100%

principal x time

Example RM 165 is interest into three shares in the

ratio 2 : 4 : 5 find the biggest share.

Solution : total ratio = 2 + 4 + 5 = 11 165 = 11 1 = 165 11 = RM 15

The biggest share is 5 = 15 x 5 = RM 75

Example 2An amount of RM 288 is shared among three

people in the ratio 1 : 3 : 5. Find the difference between the largest and the smallest share.

Solution :Total ratio = 1 + 3 + 5 = therefore 160 - 32 = 9 = RM 128

RM 288 = 9 1 = 288 9 = RM 32The largest ratio = 5 = 5 x 32 = RM 160 the smallest ratio = 1 = 1 x 32 = RM 32

Thanks you BASIC MATHEMATICS