Proportions

overview

Two ratios are said to be in proportion, if the ratios are equivalent. For example 2:42:4 and 3:63:6 are equivalent. Such equivalent ratios are formally represented as a proportion. The representation is 2:4::3:62:4::3:6.

illustrative example

A ratio of two quantities helps: to understand and to use the comparative measure of quantities.

Let us consider making dough for chappati or pizza. The recipe gives that for 400400 gram of flour, 150150ml water is used.

One person has only 200200 gram of flour. In that case, the person can 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.

proportion

The word "proportion" means: comparative measurement of quantities".

Pro-portion was from root word meaning "person's portion or share".

**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.

The word "extreme" means: farthest from the center.

The root word is from "exter" meaning outer.

The word "mean" means: average. The word is derived from a root word meaning "middle".

illustrative example

Consider two fruit baskets

• 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 the two baskets is 4:16::20:80.

*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.

The proportion of count in basket A to count in basket B for apples and oranges 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*

Consider two fruit baskets

• 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.

proportion to fractions

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 34 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.The following are all true.

the number of apples is 34 of the number of oranges in basket A

the number of apples is 68 of the number of oranges in basket B

the number of apples is 34 of the number of oranges -- in both basket A and basket B

Note that in a proportion, the two fractions are equivalent fractions.

68, when simplified, is 34.

Given the proportion 3:4::6:8,

34=68, that is, the two ratios given as fractions are always equal.

formula

Given a proportion, a:b::c:d, it is understood that ab=cd. In that case,

a×d=b×c.

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

Given that

ab=cd

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=2×2, then multiplying by 5 we get 4×5=2×2×5.)

ab and cd are two numbers that are equal. On multiplying these numbers by bd, we get the two numbers ab×bd and cd×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.*

examples

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

The answer is "9". Products of extremes equals product of means.

3×12=4×x

x=364

x=9

summary

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

**Mean-Extreme Property of Proportions** :
*product of extremes = product of means*

If a:b::c:d is a proportion, then ad=bc

Outline

The outline of material to learn "commercial arithmetics" is as follows.

Note: * Click here for the detailed ouline of commercial arthmetics.*

• **Ratio, Proportion, Percentage**

→ __Comparing Quantities__

→ __Introduction to Ratio__

→ __Ration & Fraction Differences__

→ Proportions__P__

→ __Percentages__

→ __Conversion to percentage__

• **Unitary Method**

→ __Introduction to Unitary Method__

→ __Direct Variation__

→ __Inverse Variation__

→ __DIV Pair__

• **Simple & Compound Interest**

→ __Story of Interest__

→ __Simple Interest__

→ __Compound Interest__

• **Rate•Span=Aggregate**

→ __Understanding Rate-Span__

→ __Speed • Time=Distance__

→ __Work-rate • time = Work-amount__

→ __Fill-rate • time = Filled-amount__

• **Profit-Loss-Discount-Tax**

→ __Profit-Loss__

→ __Discount__

→ __Tax__

→ __Formulas__