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Ratio & Partnership
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CAT 2025 Lesson : Ratio & Partnership - Comparing Ratios

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3. Comparing Ratios with 2 terms

Ratios with 22 terms can be expressed as a fraction. Therefore, only these ratios can be measured and compared.

By comparing these fractions, we can ascertain if the ratios are equal or if one is greater than the other. There are two ways to determine this – 3.13.1 Cross-multiplication or LCM method and 3.23.2 Percentage Comparison method.

3.1 Cross-multiplication or LCM method

If ac=cd\dfrac{a}{c} = \dfrac{c}{d}, then ad=bcad = bc

If ab>cd\dfrac{a}{b} \gt \dfrac{c}{d}, then ad>bcad \gt bc

If ab<cd\dfrac{a}{b} \lt \dfrac{c}{d}, then ad<bcad \lt bc

Therefore, we can cross-multiply the numerators with the denominators and then compare.

Example 5

If a:b=9:34a : b = 9 : 34 and c:d=12:43c : d = 12 : 43, which ratio is greater?

Solution

ab=934\dfrac{a}{b} = \dfrac{9}{34} and cd=1243\dfrac{c}{d} = \dfrac{12}{43}

a×d=9×43a \times d = 9 \times 43 and c×b=12×34c \times b = 12 \times 34

9×43=3879 \times 43 = 387 and 12×34=40812 \times 34 = 408

387<408387 \lt 408

∴ 934<1243\dfrac{9}{34} \lt \dfrac{12}{43}

Answer: c:dc : d is greater

3.2 Percentage Comparison method

If the ratio ab\dfrac{a}{b} is multiplied with mn\dfrac{m}{n},

then ambn>ab\dfrac{am}{bn} \gt \dfrac{a}{b} where m>nm \gt n;

and ambn<ab\dfrac{am}{bn} \lt \dfrac{a}{b} where m<nm \lt n

Example 6

If a:b=9:34a : b = 9 : 34 and c:d=12:43c : d = 12 : 43, which ratio is greater?

Solution

ab=934\dfrac{a}{b} = \dfrac{9}{34} and cd=1243\dfrac{c}{d} = \dfrac{12}{43}

Comparing the numerators, we note that 1212 is one-third or 33.3%33.3 \% more than 99.

Comparing the denominators,
One-third of 34=343=11.3334 = \dfrac{34}{3} = 11.33

One-third more than 34=34+11.33=45.3334 = 34 + 11.33 = 45.33

∴ 934=1245.33\dfrac{9}{34} = \dfrac{12}{45.33}

[Note: Higher the denominator, lower the fraction.]

∴ 1245.33<1243\dfrac{12}{45.33} \lt \dfrac{12}{43} ⇒ ab<cd\dfrac{a}{b} \lt \dfrac{c}{d}

Answer: c:dc : d is greater

Percentage comparison is useful when comparing fractions or ratios that are either quite far apart in value or where the percentage computations are easy.

For instance, let's take two ratios 58\dfrac{5}{8} and 712\dfrac{7}{12}.

In the 2nd2^{\text{nd}} ratio, the numerator 77 is 40%40 \% more than 55, while the denominator 1212 is 50%50 \% higher than 88.

As the denominator has increased by a higher percentage, the 2nd2^{\text{nd}} ratio is smaller than the first, i.e. 58>712\dfrac{5}{8} \gt \dfrac{7}{12}

4. Changes to ratio when a constant is added or subtracted

The following apply where a,ba, b and kk are positive numbers and arithmetic operations, such as (a+k),(a−k),(b+k)(a + k), (a - k), (b + k) and (b−k)(b - k) are also positive.

11) If ab>1\dfrac{a}{b} \gt 1 and k>0k \gt 0, then (a+k)(b+k)<ab\dfrac{(a + k)}{(b + k)} \lt \dfrac{a}{b} and (a−k)(b−k)>ab\dfrac{(a - k)}{(b - k)} \gt \dfrac{a}{b}

22) If ab<1\dfrac{a}{b} \lt 1 and k>0k \gt 0, then (a+k)(b+k)>ab\dfrac{(a + k)}{(b + k)} \gt \dfrac{a}{b} and (a−k)(b−k)<ab\dfrac{(a - k)}{(b - k)} \lt \dfrac{a}{b}

For instance, 52=2.5,(5+1)(2+1)=2<2.5\dfrac{5}{2} = 2.5, \dfrac{(5 + 1)}{(2 + 1)} = 2 \lt 2.5 and (5−1)(2−1)=4>2.5\dfrac{(5 - 1)}{(2 - 1)} = 4 \gt 2.5

For instance, 23=0.67,(2+1)(3+1)=0.75>0.67\dfrac{2}{3} = 0.67, \dfrac{(2 + 1)}{(3 + 1)} = 0.75 \gt 0.67 and (2−1)(3−1)=0.5<0.67\dfrac{(2 - 1)}{(3 - 1)} = 0.5 \lt 0.67

In case you have difficulties remembering the above conditions, you can use a simple fraction (like 52\dfrac{5}{2} or 23\dfrac{2}{3} shown above), to substitute and check.

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