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Arithmetic I

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Ratio & Partnership

Ratio And Partnership

MODULES

Basics of Ratio & Shortcuts
Properties of Ratio
Comparing Ratios
Combining, Using Variables & Multiplying
Partnership
Past Questions

CONCEPTS & CHEATSHEET

Concept Revision Video

SPEED CONCEPTS

Ratio and Proportion 1
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Ratio and Proportion 2
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Ratio and Proportion 3
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PRACTICE

Ratio & Partnership : Level 1
Ratio & Partnership : Level 2
Ratio & Partnership : Level 3
ALL MODULES

CAT 2025 Lesson : Ratio & Partnership - Comparing Ratios

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3. Comparing Ratios with 2 terms

Ratios with 222 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 – 3.13.13.1 Cross-multiplication or LCM method and 3.23.23.2 Percentage Comparison method.

3.1 Cross-multiplication or LCM method

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

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

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

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

Example 5

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

Solution

ab=934\dfrac{a}{b} = \dfrac{9}{34}ba​=349​ and cd=1243\dfrac{c}{d} = \dfrac{12}{43}dc​=4312​

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

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

387<408387 \lt 408387<408

∴ 934<1243\dfrac{9}{34} \lt \dfrac{12}{43}349​<4312​

Answer: c:dc : dc:d is greater

3.2 Percentage Comparison method

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

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

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

Example 6

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

Solution

ab=934\dfrac{a}{b} = \dfrac{9}{34}ba​=349​ and cd=1243\dfrac{c}{d} = \dfrac{12}{43}dc​=4312​

Comparing the numerators, we note that 121212 is one-third or 33.3%33.3 \%33.3% more than 999.

Comparing the denominators,
One-third of 34=343=11.3334 = \dfrac{34}{3} = 11.3334=334​=11.33

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

∴ 934=1245.33\dfrac{9}{34} = \dfrac{12}{45.33}349​=45.3312​

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

∴ 1245.33<1243\dfrac{12}{45.33} \lt \dfrac{12}{43}45.3312​<4312​ ⇒ ab<cd\dfrac{a}{b} \lt \dfrac{c}{d}ba​<dc​

Answer: c:dc : 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}85​ and 712\dfrac{7}{12}127​.

In the 2nd2^{\text{nd}}2nd ratio, the numerator 777 is 40%40 \%40% more than 555, while the denominator 121212 is 50%50 \%50% higher than 888.

As the denominator has increased by a higher percentage, the 2nd2^{\text{nd}}2nd ratio is smaller than the first, i.e. 58>712\dfrac{5}{8} \gt \dfrac{7}{12}85​>127​

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

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

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

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

For instance, 52=2.5,(5+1)(2+1)=2<2.5\dfrac{5}{2} = 2.5, \dfrac{(5 + 1)}{(2 + 1)} = 2 \lt 2.525​=2.5,(2+1)(5+1)​=2<2.5 and (5−1)(2−1)=4>2.5\dfrac{(5 - 1)}{(2 - 1)} = 4 \gt 2.5(2−1)(5−1)​=4>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 0.6732​=0.67,(3+1)(2+1)​=0.75>0.67 and (2−1)(3−1)=0.5<0.67\dfrac{(2 - 1)}{(3 - 1)} = 0.5 \lt 0.67(3−1)(2−1)​=0.5<0.67

In case you have difficulties remembering the above conditions, you can use a simple fraction (like 52\dfrac{5}{2}25​ or 23\dfrac{2}{3}32​ shown above), to substitute and check.

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