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Thought-Process to Discover Knowledge

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mathsCommercial ArithmeticsComparing Quantities

### Proportions

Two ratios are said to be in proportion, if the ratios are equivalent. For example 2:4 and 3:6 are equivalent. Such equivalent ratios are formally represented as a proportion. The representation is 2:4::3:6.

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What is the use of a ratio?

• to understand and to use the comparative measure of quantities
• to understand and to use the comparative measure of quantities
• there is no practical use

The answer is "to understand and to use the comparative measure of quantities"

Let us consider making dough for chappati or pizza. The recipe gives that for 400 gram of flour, 150ml water is used.

One person has only 200 gram of flour. What shall he do?

• buy another 200 gram of flour to mix with 150 ml water
• reduce the water to 75ml
• reduce the water to 75ml

The answer is "reduce the water to 75ml".

For 400 gram flour 150ml water is used. This is in 400:150 ratio.

For 200 gram flour 75ml water is used. This is in 200:75 ratio.

These two ratios can be simplified to 8:3.

To denote that two ratios are identical, they are said to be in same proportion. That is 400:150 is in the same proportion as 200:75. This is given as 400:150::200:75.

Which of the following is a meaning for the word "proportion"?

• comparative measurement of quantities
• comparative measurement of quantities
• professional player taking part in only few matches

The answer is "comparative measurement of quantities". Pro-portion was from root word meaning "person's portion or share".

What is the term used to refer "two ratios being equal when simplified"?

• Pronunciation : Say the answer once

Proportion : Two ratios are said to be in proportion if the corresponding terms of ratio are identical in the simplified form.

Consider the example 2:3::4:6 proportion.

•  The numbers in the proportion are called first term, second term, third term, and fourth term in the order.

•  The first and fourth terms are called the extremes of the proportion.

•  The second and third terms are called the means of the proportion.

Which of the following is a meaning for the word "extreme"?

• farthest from the center
• farthest from the center
• a way of living

The answer is "farthest from the center". The root word is from "exter" meaning outer.

Which of the following is a meaning for the word "mean"?

• average
• average
• a type of fish

The answer is "average". The word is derived from a root word meaning "middle".

What is the term used to refer "condition or quality that is removed from far extremes"?

• Pronunciation : Say the answer once

What is the term used to refer "farthest from the center"?

• Pronunciation : Say the answer once

•  Fruit basket A has 4 apples and 16 oranges. Ratio of apples to oranges is 4:16 which is 1:4.

•  Fruit basket B has 20 apples and 80 oranges. Ratio of apples to oranges is 20:80 which is 1:4.

What is the proportion of apples to oranges in the two baskets?

• 4:16::20:80
• 4:16::20:80
• 4:20::16:80

The answer is "4:16::20:80".

•  Fruit basket A has 4 apples and 16 oranges. Ratio of apples to oranges is 4:16 which is 1:4.

•  Fruit basket B has 20 apples and 80 oranges. Ratio of apples to oranges is 20:80 which is 1:4.

The proportion of apples to oranges in basket A and basket B is 4:16::20:80.

The word "Proportion" is also used to specify the simplified ratio, as in the following.

The proportion of apples to oranges in basket A and basket B is 1:4.

Students are reminded to note the context in which the word "proportion" is used.

•  Fruit basket A has 4 apples and 16 oranges. Ratio of apples to oranges is 4:16 which is 1:4.

•  Fruit basket B has 20 apples and 80 oranges. Ratio of apples to oranges is 20:80 which is 1:4.

What is the proportion of count in basket A to count in basket B for apples and oranges?

• 4:16::20:80
• 4:20::16:80
• 4:20::16:80

The answer is "4:20::16:80". This proportion is given as

apples of basket A to basket B is in the same proportion as oranges of basket A to basket B

•  Fruit basket A has 4 apples and 16 oranges.

•  Fruit basket B has 20 apples and 80 oranges.

The proportion of apples to oranges in basket A and basket B is 4:16::20:80.

The proportion of apples and oranges in basket A to basket B is 4:20::16:80.

Students are reminded to note the context in which proportion is defined.

A basket has 3 apples and 4 oranges.

What is the ratio of number of apples to oranges?

• 3:4
• 3:4
• 3:7

The answer is "3:4"

A basket has 3 apples and 4 oranges. The ratio of the number of apples to number of oranges is 3:4.

How many apples are there in comparison to the number of oranges?

• Number of apples is 3/4 of the number of oranges.
• Number of apples is 3/4 of the number of oranges.
• Number of apples is 3/7 of the number of oranges

The answer is "Number of apples is 3/4 of the number of oranges."

We learned that "Ratio can be equivalently represented as a fraction.".

A basket has 3 apples and 4 oranges.

•  The ratio of the number of apples to number of oranges is 3:4.

•  Number of apples are 3/4 of the number of oranges.

The number of apples to number of oranges in basket A and B is in proportion 3:4::6:8. Which of the following is correct?

• the number of apples is 3/4 of the number of oranges in basket A
• the number of apples is 6/8 of the number of oranges in basket B
• the number of apples is 3/4 of the number of oranges in both basket A and basket B
• all the above
• all the above

The answer is "all the above".

Ratio can be equivalently represented as a fraction.
eg: The number of apples and number of oranges are in 3:4 ratio.
The number of apples are 3/4 of the number of oranges.

Similarly, Proportion can also be equivalently represented as a fraction.
eg: The number of apples to number of oranges in basket A and B is in proportion 3:4::6:8.
The number of apples is 3/4 of the number of oranges in basket A and 6/8 in basket B.

Note that in a proportion, the two fractions are equivalent fractions.
6/8, when simplified, is 3/4.

Given the proportion 3:4::6:8, which of the following is correct?

• 3/4 = 6/8, that is, the two ratios given as fractions are always equal
• 3/4 = 6/8, that is, the two ratios given as fractions are always equal
• The two ratios given as fractions need not be equal for all proportions

The answer is "the two ratios given as fractions are always equal"

Given a proportion, a:b::c:d, it is understood that a/b = c/d.

In that case, which of the following is correct?

• a xx b = c xx d
• a xx d = b xx c
• a xx d = b xx c

The answer is "a xx d = b xx c". This is explained in the next page.

Given a proportion, a:b::c:d, it is understood that a/b = c/d.

Given that
a/b = c/d

Note that a, b, c, d are numbers. As per the properties of numbers, if two numbers are equal, then the numbers multiplied by another number are equal. (eg: if  4=2 xx 2, then multiplying by 5 we get 4xx5 = 2xx2xx5.)
a/b and c/d are two numbers that are equal. On multiplying these numbers by bd, we get the two numbers a/b xx bd and c/d xx bd. Simplifying these two numbers we get, ad and bc. As per the property, these two numbers are equal.

ad = bc.

That is product of extremes and product of means are equal.

Mean-Extreme Property of Proportions : product of extremes = product of means
If a:b::c:d is a proportion, then ad = bc

Solved Exercise Problem:

Given the proportion 3:4::x::12 find the value of x.

• 9
• 9
• 3

The answer is "9". Products of extremes equals product of means.
3 xx 12 = 4 xx x
x = 36/4
x = 9

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