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Arithmetic I

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Ratio & Partnership
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CAT 2025 Lesson : Ratio & Partnership - Combining, Using Variables & Multiplying

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5. Combining Ratios

5.1 Combining two ratios

If the ratios of a:ba : b and b:cb : c are given and we need to find out a:b:ca : 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=3:7a : b = 3 : 7 and b:c=2:5b : c = 2 : 5, then a:b:c=a : b : c = ?

Solution

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

ab=37=3×27×2=614=6:14\dfrac{a}{b} = \dfrac{3}{7} = \dfrac{3 \times 2}{7 \times 2} = \dfrac{6}{14} = 6 : 14

bc=25=2×75×7=1435=14:35\dfrac{b}{c} = \dfrac{2}{5} = \dfrac{2 \times 7}{5 \times 7} = \dfrac{14}{35} = 14 : 35

For every 66 parts of aa, there are 1414 parts of bb. And, for every 1414 parts of bb, there are 3535 parts of cc.

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.

                    a       :       b       :       c\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 11         3       :       7\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 22                             2       :       5\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 11 is multiplied by 22 and ratio 22 is multiplied by 77, the b term will be common.

                        a          :       b       :       c\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 11         3×2       :     7×2\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 22                                 2×7    :    5×7\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: 6:14:356 : 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 ab=p1q1,bc=p2q2\dfrac{a}{b} = \dfrac{p_{1}}{q_{1}}, \dfrac{b}{c} = \dfrac{p_{2}}{q_{2}}, then a:b:c=p1p2:q1p2:q1q2a : b : c = p_{1}p_{2} : q_{1}p_{2} : q_{1}q_{2}

Working Example 77 in this method, where a:b=3:7a : b = 3 : 7 and b:c=2:5b : c = 2 : 5
⇒ a:b:c=(3×2):(7×2):(7×5)a : b : c = (3 \times 2) : (7 \times 2) : (7 \times 5) =6:14:35= 6 : 14 : 35

If ab=p1q1,bc=p2q2,cd=p3q3\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=p1p2p3:q1p2p3:q1q2p3:q1q2q3a : 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 ab=p1q1,bc=p2q2,\dfrac{a}{b} = \dfrac{p_{1}}{q_{1}}, \dfrac{b}{c} = \dfrac{p_{2}}{q_{2}}, cd=p3q3,de=p4q4\dfrac{c}{d} = \dfrac{p_{3}}{q_{3}}, \dfrac{d}{e} = \dfrac{p_{4}}{q_{4}}, then

a:b:c:d:e=p1p2p3p4:q1p2p3p4:q1q2p3p4:q1q2q3p4:q1q2q3q4a : 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, 11 denominator replaces 11 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 ab=25\dfrac{a}{b} = \dfrac{2}{5}, cb=73\dfrac{c}{b} = \dfrac{7}{3}, cd=23\dfrac{c}{d} = \dfrac{2}{3}, ed=59\dfrac{e}{d} = \dfrac{5}{9} and aa & ee are the smallest possible natural numbers, then a+e=?a + e = ?

Solution

We rewrite them in order as ab=25\dfrac{a}{b} = \dfrac{2}{5}, bc=37\dfrac{b}{c} = \dfrac{3}{7}, cd=23\dfrac{c}{d} = \dfrac{2}{3}, de=95\dfrac{d}{e} = \dfrac{9}{5}

ae=(2×3×2×9)(5×7×3×5)=36175\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=211a + e = 36 + 175 = 211

Answer: 211211

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 3:5:73 : 5 : 7, then 3x,5x3x, 5x and 7x7x can be taken as their respective values. Note that x\bm{x} is the only variable here.

Example 9

22 years back the ratio of Rachel and her mother's ages was 1:31 : 3. 8 years from now, the ratio of their ages will be 1:21 : 2. What is the ratio of their present ages?

Solution

Let the ages of Rachel and her mother 22 years back be xx and 3x3x respectively.

Their respective ages 88 years from now will be x+10x + 10 and 3x+103x + 10. These are said to be in the ratio 1:21 : 2.

∴ x+103x+10=12\dfrac{x + 10}{3x + 10} = \dfrac{1}{2}

⇒ 2x+20=3x+102x + 20 = 3x + 10 ⇒ x=10 x = 10

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

Answer: 3:83 : 8

6.2 Multiplying/Dividing Ratios

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

For instance,Distance = Speed ×\times Time. Therefore, if the ratio of speeds of three people is 2:3:52 : 3 : 5 and the ratio of their time-taken is 1:2:31 : 2 : 3, then the ratio of the distance covered by them is (2×1):(3×2):(5×3)=2:6:15(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 20202020 is 5:4:35 : 4 : 3, while the ratio of quantities sold is 3:4:53 : 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= 5 : 4 : 3, let the Revenue earned from X, Y and Z be 5a,4a5a, 4a and 3a3a respectively.
As Ratio of Quantity Sold =3:4:5= 3 : 4 : 5, let the Quantity of X, Y and Z sold be 3b,4b3b, 4b and 5b5b respectively.

Unit Price of X, Y and Z are 5a3b,4a4b\dfrac{5a}{3b}, \dfrac{4a}{4b} and 3a5b\dfrac{3a}{5b} respectively.

Ratio of unit prices =5a3b:4a4b:3a5b=53:1:35= \dfrac{5a}{3b} : \dfrac{4a}{4b} : \dfrac{3a}{5b} = \dfrac{5}{3} : 1 : \dfrac{3}{5}

Multiplying with the LCM (3,5)=15(3, 5) = 15,
=53×15:1×15:35×15= \dfrac{5}{3} \times 15 : 1 \times 15 : \dfrac{3}{5} \times 15

= 25 : 15 : 9

Alternatively (Recommended Method)

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

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

Ratio of Unit Price =53:44:35= \dfrac{5}{3} : \dfrac{4}{4} : \dfrac{3}{5}

= 25 : 15 : 9

Answer: 25:15:925 : 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 55 apples, 66 mangoes and 88 guavas is 5:10:45 : 10 : 4, then what is the ratio of the prices of 44 apples, 33 mangoes and 66 guavas?

(1) 3:2:13 : 2 : 1            (2) 4:2:34 : 2 : 3            (3) 3:5:23 : 5 : 2            (4) 4:5:34 : 5 : 3           

Solution

Ratio of 55 apples, 66 mangoes and 88 guavas =5:10:4= 5 : 10 : 4

Ratio of 11 apple, 11 mango and 11 guava =55:106:48=1:53:12=6:10:3= \dfrac{5}{5} : \dfrac{10}{6} : \dfrac{4}{8} = 1 : \dfrac{5}{3} : \dfrac{1}{2} = 6 : 10 : 3

Ratio of 44 apples, 33 mangoes and 66 guavas =4×6:3×10:6×3=24:30:18=4:5:3= 4 \times 6 : 3 \times 10 : 6 \times 3 = 24 : 30 : 18 = 4 : 5 : 3

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

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