Commercial Arithmetics : Proportions

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Proportions

what you'll learn...

overview

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

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 $400$ $400$ gram of flour, $150$ $150$ ml water is used.

One person has only $200$ $200$ gram of flour. In that case, the person can reduce the water to $75$ ml.

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

For $200$ gram flour $75$ ml 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 $3/4$ of the number of oranges.

The number of apples to number of oranges in basket A and B is in proportion $3:4::6:8$ .The following are all true.

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

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

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

$3/4 = 6/8$ , that is, the two ratios given as fractions are always equal.

formula

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

$a xx d = b xx c$ .

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.

examples

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

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

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

→ ProportionsP

→ 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