Concept of Ratio, Proportion and Variation

Ratio:

While comparing any two numbers, sometimes, it is necessary to find out how many times one number is greater or less than other. It is denoted by $$\frac{{\text{x}}}{{\text{y}}}$$ or x : y. The numbers that forms the ratio are called the terms of the ratio. The numerator of the ratio is called antecedent and denominator is called the consequent of the ratio.
Ratio may be taken for homogeneous quantities or for heterogeneous quantities. In the first case, the ratio has no unit, while in second case; the unit of the ratio is based on the units of the numerator and that of the denominator.

Relation between Ratio and Percentage

Ratio can be expressed as percentage. To express the value of a ratio as percentage, we multiply the ratio by 100.
Thus, $$\frac{4}{5} = \frac{4}{5} \times 100 = 80\% $$

Key Facts:

• If we multiply or divide the numerator and denominator of a ratio with the same number, the ratio remains unchanged.
That is, $$\frac{4}{5} = \frac{{4 \times 8}}{{5 \times 8}}$$
Or, $$\frac{4}{5} = \frac{{4/3}}{{5/3}}$$

• If the ratio between two corresponding sides of two figures (two dimensional) is a : b then ratio of their areas is a² : b².
Example: Ratio between two diagonal of two squares is 2 : 1. What is the ratio of their areas?

Solution:
Ratio of areas = 2² : 1² = 4 : 1.
If the ratio between two corresponding sides of two figures (three dimensional) is a : b then ratio of their volumes is a³ : b³.

Comparisons of Ratio

1. The Cross Multiplication Method:
Which one of the two ratios is greater ?
5 : 7 or 9 : 11.

Solution:
1^st ratio = $$\frac{5}{7}$$ and 2^nd ratio = $$\frac{9}{11}$$.
For comparing, denominator of both should be made equal,
$$\eqalign{ & {\text{Or,}}\,\frac{5}{7}\,{\text{or}}\,\frac{9}{{11}} \cr & {\text{Or,}}\,\frac{{5 \times 11}}{{7 \times 11}}\,{\text{or}}\,\frac{{9 \times 7}}{{11 \times 7}} \cr & {\text{Or,}}\,\frac{{55}}{{77}}\,{\text{or}}\,\frac{{63}}{{77}} \cr & \frac{{63}}{{77}}\,{\text{is}}\,{\text{greater}}\,{\text{than}}\,\frac{{55}}{{77}} \cr & {\text{Hence,}}\,\frac{9}{{11}} > \frac{5}{7} \cr} $$

2. Percentage value comparison method:-
Compare $$\frac{{173}}{{212}}$$ with $$\frac{{181}}{{241}}$$.
In such case we need to estimate the 10% ranges for each.
To find 10% range, we just need to put (.) after unit digit in denominator i.e. 10% of 212 = 21.2
Now, we have to notice 173 is roughly more than 80% of 212.$$\left[ {173 \times \frac{{100}}{{21.2}} = 80\% } \right]$$
While second ratio is < 80%.
Hence, first ratio is greater than second ratio.
However, this problem will become slightly more difficult, if the two ratios fall in the same 10% range. Thus, if you had to compare $$\frac{{173}}{{212}}$$ with $$\frac{{181}}{{225}}$$, both values would give values between 80 - 90%. The next step would be to calculate the 1% range. Then,
The first ratio is 81 - 82% while the second ratio lies between 80 - 81%. Hence, the first ratio is the larger of the two.

3. Numerator to denominator percentage change:
Suppose, we have two ratios $$\frac{{173}}{{181}}$$ and $$\frac{{181}}{{225}}$$ to compare. In this method, we need to calculate percentage changes in numerator to denominator.
173 → 181 (4-5% increase.) [You can see, $$8 \times \frac{{100}}{{173}}$$  = 4.62%, i.e. change lies between 4-5%]
While, 212 → 225 (6-7% increase.) $$\left[ {13 \times \frac{{100}}{{212}} = 6.13\% } \right]$$
Since, in second case increase in numerator to denominator is more so second ratio is smaller. i.e. $$\frac{{212}}{{225}} < \frac{{173}}{{181}}$$

Calculation of percentage change in the ratios:

Suppose, we have to calculate the percentage change of 2 ratios. This can be done in two steps as:
Original Ratio ------ (numerator effects) --> Intermediate ratio
---------- (Denominator effects) ----> Final ratio.
Suppose, $$\frac{{20}}{{40}}$$ becomes $$\frac{{22}}{{50}}$$
Numerator effect = 20 → 22 (10% increase).
Denominator effect = 50 → 40 (25 increase). [we take it in reverse way as the denominator increases then numerator decreases.]
Now,
100----10%↑--->110-----25%↓----> 82.5.
Hence, overall effect= 17.5 decrease.

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