5. Combining Ratios

5.1 Combining two ratios

If the ratios of $a : b$ and $b : c$ are given and we need to find out $a : b : c$, then we will have to scale up/down the two ratios such that the value of b is common, and then proceed to combine the ratios.

Example 7

If $a : b = 3 : 7$ and $b : c = 2 : 5$, then $a : b : c =$ ?

Solution

The LCM of the $b$ terms is LCM $(7, 2) = 14$

$\dfrac{a}{b} = \dfrac{3}{7} = \dfrac{3 \times 2}{7 \times 2} = \dfrac{6}{14} = 6 : 14$

$\dfrac{b}{c} = \dfrac{2}{5} = \dfrac{2 \times 7}{5 \times 7} = \dfrac{14}{35} = 14 : 35$

For every $6$ parts of $a$, there are $14$ parts of $b$. And, for every $14$ parts of $b$, there are $35$ parts of $c$.

Now the ratios can be combined as 6 : 14 : 35.

Alternatively

We write the ratios one below the other, in such a way that the terms in the ratios are under their corresponding variables.

$\space \space \space \space \space \space \space \space \space\space \space \space \space \space \space \space \space \space \space \space \bm{a} \space \space \space \space \space \space \space : \space \space \space \space\space \space \space \bm{b} \space \space \space \space \space \space \space : \space \space \space \space \space \space \space \bm{c}$
Ratio $1$ $\space \space \space \space \space \space \space \space 3 \space \space \space \space \space \space \space : \space \space \space \space \space \space \space 7$
Ratio $2$ $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space\space \space \space 2 \space \space \space \space \space \space \space : \space \space \space \space \space \space \space 5$

If ratio $1$ is multiplied by $2$ and ratio $2$ is multiplied by $7$, the b term will be common.

$\space \space \space \space \space \space \space \space \space \space \space \space \space\space \space \space \space \space \space \space \space \space \space \space \bm{a} \space \space \space \space \space \space \space \space \space \space : \space \space \space \space\space \space \space \bm{b} \space \space \space \space \space \space \space : \space \space \space \space \space \space \space \bm{c}$
Ratio $1$ $\space\space \space \space \space \space \space \space 3 \times 2 \space \space \space \space \space \space \space : \space \space \space \space \space 7 \times 2$
Ratio $2$ $\space\space\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space\space \space \space 2 \times 7 \space \space \space \space : \space \space \space \space 5 \times 7$

Ratios can now be combined as 6 : 14 : 35.

Answer: $6 : 14 : 35$

5.2 Combining multiple ratios

Where multiple ratios are to be combined, combining ratios one by one (by scaling up or down) becomes cumbersome. Here, the following method can be applied.

If $\dfrac{a}{b} = \dfrac{p_{1}}{q_{1}}, \dfrac{b}{c} = \dfrac{p_{2}}{q_{2}}$, then $a : b : c = p_{1}p_{2} : q_{1}p_{2} : q_{1}q_{2}$

Working Example $7$ in this method, where $a : b = 3 : 7$ and $b : c = 2 : 5$
⇒ $a : b : c = (3 \times 2) : (7 \times 2) : (7 \times 5)$ $= 6 : 14 : 35$

If $\dfrac{a}{b} = \dfrac{p_1}{q_{1}}, \dfrac{b}{c} = \dfrac{p_{2}}{q_{2}}, \dfrac{c}{d} = \dfrac{p_{3}}{q_{3}}$, then $a : b : c : d = p_{1}p_{2}p_{3} : q_{1}p_{2}p_{3} : q_{1}q_{2}p_{3} : q_{1}q_{2}q_{3}$

If $\dfrac{a}{b} = \dfrac{p_{1}}{q_{1}}, \dfrac{b}{c} = \dfrac{p_{2}}{q_{2}},$ $\dfrac{c}{d} = \dfrac{p_{3}}{q_{3}}, \dfrac{d}{e} = \dfrac{p_{4}}{q_{4}}$, then

$a : b : c : d : e = p_{1}p_{2}p_{3} p_{4} : q_{1}p_{2}p_{3} p_{4} : q_{1}q_{2}p_{3} p_{4} : q_{1}q_{2}q_{3} p_{4} : q_{1}q_{2}q_{3} q_{4}$

Note that the first term is the product of all the numerators. In every subsequent term, $1$ denominator replaces $1$ numerator. This results in the product of all the denominators as the last term. It is easy to remember it this way.

Example 8

If $\dfrac{a}{b} = \dfrac{2}{5}$, $\dfrac{c}{b} = \dfrac{7}{3}$, $\dfrac{c}{d} = \dfrac{2}{3}$, $\dfrac{e}{d} = \dfrac{5}{9}$ and $a$ & $e$ are the smallest possible natural numbers, then $a + e = ?$

Solution

We rewrite them in order as $\dfrac{a}{b} = \dfrac{2}{5}$, $\dfrac{b}{c} = \dfrac{3}{7}$, $\dfrac{c}{d} = \dfrac{2}{3}$, $\dfrac{d}{e} = \dfrac{9}{5}$

$\dfrac{a}{e} = \dfrac{(2 \times 3 \times 2 \times 9)}{(5 \times 7 \times 3 \times 5)} = \dfrac{\bm{36}}{\bm{175}}$

$a + e = 36 + 175 = 211$

Answer: $211$

6. Application of Ratios

6.1 Using variables

In questions of this type, we will be given the ratios and asked to find the absolute values.

The quantities linked through a ratio can be represented with only 1 variable. For instance, if three quantities are in the ratio $3 : 5 : 7$, then $3x, 5x$ and $7x$ can be taken as their respective values. Note that $\bm{x}$ is the only variable here.

Example 9

$2$ years back the ratio of Rachel and her mother's ages was $1 : 3$. 8 years from now, the ratio of their ages will be $1 : 2$. What is the ratio of their present ages?

Solution

Let the ages of Rachel and her mother $2$ years back be $x$ and $3x$ respectively.

Their respective ages $8$ years from now will be $x + 10$ and $3x + 10$. These are said to be in the ratio $1 : 2$.

∴ $\dfrac{x + 10}{3x + 10} = \dfrac{1}{2}$

⇒ $2x + 20 = 3x + 10$ ⇒ $ x = 10$

Ratio of their present ages is $(x + 2) : (3x + 2) = 12 : 32 = 3 : 8$

Answer: $3 : 8$

6.2 Multiplying/Dividing Ratios

When $a \times b = c$, and if ratio of $a$-values and $b$-values are given, then to find the ratio of the $c$-values, we can directly multiply the corresponding $a$-terms and $b$-terms in the respective ratios.

For instance,Distance = Speed $\times$ Time. Therefore, if the ratio of speeds of three people is $2 : 3 : 5$ and the ratio of their time-taken is $1 : 2 : 3$, then the ratio of the distance covered by them is $(2 \times 1) : (3 \times 2) : (5 \times 3) = 2 : 6 : 15$.

The same applies for division.

Note: In case of addition or subtraction, you need to use the method outlined in 6.1 Using Variables.

Example 10

Company ABC sold 3 products – X, Y and Z. The ratio of the revenue earned from X, Y and Z in $2020$ is $5 : 4 : 3$, while the ratio of quantities sold is $3 : 4 : 5$. If the unit prices for the respective products remained constant throughout the year, what was the ratio of their unit prices?

Solution

Using Variables
As Ratio of Revenues $= 5 : 4 : 3$, let the Revenue earned from X, Y and Z be $5a, 4a$ and $3a$ respectively.
As Ratio of Quantity Sold $= 3 : 4 : 5$, let the Quantity of X, Y and Z sold be $3b, 4b$ and $5b$ respectively.

Unit Price of X, Y and Z are $\dfrac{5a}{3b}, \dfrac{4a}{4b}$ and $\dfrac{3a}{5b}$ respectively.

Ratio of unit prices $= \dfrac{5a}{3b} : \dfrac{4a}{4b} : \dfrac{3a}{5b} = \dfrac{5}{3} : 1 : \dfrac{3}{5}$

Multiplying with the LCM $(3, 5) = 15$,
$= \dfrac{5}{3} \times 15 : 1 \times 15 : \dfrac{3}{5} \times 15$

= 25 : 15 : 9

Alternatively (Recommended Method)

Unit Price $= \dfrac{\text{Revenue}}{\text{Quantity Sold}}$

Ratio of Revenue $= 5 : 4 : 3$
Ratio of Quantity sold $= 3 : 4 : 5$

Ratio of Unit Price $= \dfrac{5}{3} : \dfrac{4}{4} : \dfrac{3}{5} $

= 25 : 15 : 9

Answer: $25 : 15 : 9$

We can also multiply specific terms of a ratio to get the desired ratio.

Example 11

If the ratio of the prices of $5$ apples, $6$ mangoes and $8$ guavas is $5 : 10 : 4$, then what is the ratio of the prices of $4$ apples, $3$ mangoes and $6$ guavas?

(1) $3 : 2 : 1$ (2) $4 : 2 : 3$ (3) $3 : 5 : 2$ (4) $4 : 5 : 3$

Solution

Ratio of $5$ apples, $6$ mangoes and $8$ guavas $= 5 : 10 : 4$

Ratio of $1$ apple, $1$ mango and $1$ guava $= \dfrac{5}{5} : \dfrac{10}{6} : \dfrac{4}{8} = 1 : \dfrac{5}{3} : \dfrac{1}{2} = 6 : 10 : 3$

Ratio of $4$ apples, $3$ mangoes and $6$ guavas $= 4 \times 6 : 3 \times 10 : 6 \times 3 = 24 : 30 : 18 = 4 : 5 : 3$

Answer: $(4) 4 : 5 : 3$