Comparing Ratios: Formulas, Methods and Examples

Comparing ratios means deciding whether one ratio is greater than, less than or equal to another. The comparison can be made by converting ratios to equivalent forms, using cross multiplication, or expressing them as fractions or decimals. This page covers the comparing ratios formula, direct methods, shortcuts and solved examples for quantitative aptitude questions.

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What is Comparing Ratios?

Comparing ratios is the process of determining the relationship between two ratios, such as 3:5 and 4:7. For positive ratios a:b and c:d, compare the products ad and bc.

Write the ratios as fractions: a:b = a/b and c:d = c/d. If ad > bc, then a:b > c:d; if ad < bc, then a:b < c:d; and if ad = bc, the ratios are equal. The quantities compared must use the same units. For example, 2:3 and 5:8 give 2 × 8 = 16 and 3 × 5 = 15, so 2:3 > 5:8.

Comparing Ratios Video Lessons

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Ratio Basics: Simplify, Compare and Scale

Learn the fundamentals of ratios by simplifying ratio expressions, comparing quantities, and scaling ratios while preserving their relationships.

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Comparing Ratios Formula & Tricks

Important Formulas

Cross-multiplication formula
For a:b and c:d, compare ad with bc.

If ad > bc, then a:b > c:d. If ad < bc, then a:b < c:d. If ad = bc, the ratios are equal.

Difference of two ratios
a/b − c/d = (ad − bc)/(bd)

For positive b and d, the sign of ad − bc directly shows which ratio is larger.

Equivalent ratio
a:b = ka:kb, where k > 0

Multiplying both terms of a ratio by the same positive number does not change its value.

Ratio as a decimal
a:b = a ÷ b

Convert each ratio into a decimal and compare the decimal values when division is convenient.

Quick Tricks

Cross-multiply without reducing first

For two positive ratios, compare the products of opposite terms. This avoids unnecessary simplification and works especially well with large numbers.

Example: To compare 17:23 and 19:26, calculate 17 × 26 = 442 and 19 × 23 = 437. Since 442 > 437, 17:23 is greater.
Make one term common

If the first terms or second terms can be made equal easily, compare the remaining terms directly.

Example: Compare 5:8 and 15:22. Multiply 5:8 by 3 to get 15:24. Since 24 > 22, 15:22 is greater than 5:8.
Use nearby benchmark ratios

When ratios are close to a simple benchmark such as 1, compare their deviations carefully. A ratio above 1 is greater than a ratio below 1.

Example: 7:6 is greater than 1, while 9:10 is less than 1. Therefore, 7:6 > 9:10.

Comparing Ratios Concepts

Comparing Ratios by Cross Multiplication

Cross multiplication gives a direct comparison of two ratios without converting them into decimals.

For a:b and c:d, form a/b and c/d. Because b and d are positive, compare ad and bc. The comparison result is unchanged when both ratios are reduced or expanded by positive factors.

Example: Compare 7:12 and 5:9. Here, 7 × 9 = 63 and 5 × 12 = 60. Since 63 > 60, 7:12 > 5:9.

Comparing Ratios by Equivalent Ratios

Equivalent ratios can be compared by multiplying one or both ratios until a corresponding term is common.

If one ratio can be changed to have the same antecedent or consequent as the other, compare the unlike terms. Multiply or divide both terms of a ratio by the same non-zero factor.

Example: Compare 4:7 and 12:20. Multiplying 4:7 by 3 gives 12:21. Since 21 > 20 for the same first term, 12:20 > 4:7.

Comparing Ratios Using Fractions or Decimals

A ratio a:b can be treated as the fraction a/b, so decimal conversion provides another comparison method.

Divide the first term by the second term in each ratio. This method is convenient when the quotients terminate or are easy to estimate. For recurring decimals, cross multiplication is usually more reliable.

Example: Compare 3:8 and 5:12. Their values are 0.375 and approximately 0.4167, respectively. Therefore, 5:12 > 3:8.

Comparing Ratios with Different Units

Convert quantities to the same unit before forming or comparing ratios.

A ratio between unlike units cannot be compared correctly until the units are converted. Use standard conversions such as 1 m = 100 cm and 1 hour = 60 minutes, then apply cross multiplication or simplification.

Example: Compare 2 m:150 cm with 3:2. Convert 2 m to 200 cm, so the first ratio is 200:150 = 4:3. Since 4:3 > 3:2 is false, the first ratio is actually less than 3:2.

Comparing Ratios Data Table

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Methods for Comparing Two Ratios

The same ratio comparison can be solved by different numerical methods.

Method Rule Example result
Cross multiplication Compare ad and bc for a:b and c:d For 2:3 and 5:8, 16 > 15, so 2:3 > 5:8
Equivalent ratios Make a corresponding term common 4:7 = 12:21; therefore 12:20 > 4:7
Decimal conversion Divide each first term by its second term 3:8 = 0.375 and 5:12 ≈ 0.4167, so 5:12 > 3:8
Unit conversion Express both quantities in the same unit first 2 m:150 cm = 200:150 = 4:3

Note: Cross multiplication is valid directly when the ratio terms used as denominators are positive.

Practice Comparing Ratios Questions

Practise published questions related to this topic.

1Which of the following numbers is not divisible by both 94 and 68?→ 2Among three partners A, B and C in a business, B's capital is equal to half the difference between three times C's capital and twice A's capital. If they make a profit of ₹25405 at the end of the year, what will be C's share?→ 3Which of the following statement is correct? I. Sum of first 7 odd numbers is more than 50. II. If n is even, then the sum of first n prime numbers is even.→ 4The price of a trouser is first increased by 20% and then decreased by 25%. What is the net percentage change in the price?→ 5What is the lowest form of the fraction 72/96 obtained by cancelling the HCF of the numerator and denominator?→ 6Leena and Kareena entered into a partnership with investment in the ratio 4:5. Meena also joined the partnership with some investment at the same time. After 3 months, Leena withdrew one-fourth and Kareena withdrew one-fifth of their respective investments. If the gain at the end of 10 months is ₹7,480, find Meena’s share (in ₹) in the profit if Leena’s share is ₹1,815.→ 7Praveen and Hemant start a business. Praveen invests ₹78,000 more than Hemant for 4 months and Hemant for 7 months. Praveen's share is ₹1,017 more than that of Hemant, out of a total profit of ₹7,119. Find the capital contributed by Praveen.→ 8P sells an item to Q at 25% profit. Q sells the same item to R at 60% loss. If R paid ₹5,166 for the item, then at what price (in ₹) is the item purchased by P?→ 9An article costing Rs. 500 is sold with a Rs. 50 discount. What is the discount percentage?→ 10The difference between the cost price and the selling price of an object is ₹480. If the profit is 20% on the cost price, find the selling price of the object.→

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Quick Revision Notes

Comparing Ratios Revision Points

Use these rules to compare ratios accurately and quickly.

  • For a:b and c:d, compare ad and bc.
  • If ad > bc, then a:b > c:d; if ad < bc, then a:b < c:d.
  • Reduce or expand a ratio only by multiplying or dividing both terms by the same non-zero factor.
  • Convert unlike units before forming or comparing ratios.
  • Decimal conversion is suitable when the quotients are easy to calculate; cross multiplication avoids rounding errors.
  • For positive ratios, a:b = c:d when ad = bc.

Related Topic Explainers

Partnership with Change in Investment → Equivalent Ratios →

Comparing Ratios FAQs

What is the formula for comparing ratios?

For a:b and c:d, compare a × d with c × b. If ad > bc, then a:b is greater; if ad < bc, it is smaller; if they are equal, the ratios are equal.

Which ratio is greater: 11:15 or 7:10?

Cross multiplication gives 11 × 10 = 110 and 7 × 15 = 105. Therefore, 11:15 > 7:10.

How can equal ratios be identified?

Ratios a:b and c:d are equal when ad = bc. For example, 6:8 and 9:12 are equal because 6 × 12 = 8 × 9 = 72.

Can ratios with different units be compared directly?

No. Convert the quantities to the same unit first. For example, 1 m:50 cm becomes 100:50 = 2:1 after converting 1 m to 100 cm.

Why is cross multiplication preferred over decimal conversion in close comparisons?

Cross multiplication gives an exact result and avoids rounding. For example, compare 13:17 and 16:21 using 13 × 21 = 273 and 16 × 17 = 272, so 13:17 is greater.

How are three ratios compared when their terms are not the same?

Compare them in pairs using cross multiplication, or convert all of them to fractions with a common denominator. For example, compare 2:3, 3:5 and 5:8 by evaluating 2/3, 3/5 and 5/8.

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