Ratio and Percentage: Formulas, Conversions and Questions

Ratio and Percentage connect a comparison between quantities with a value out of 100. This topic covers converting ratios into percentages, changing percentages into ratios, finding each part as a percentage of the total, and solving questions involving percentage-based changes in ratios.

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What Is the Connection Between Ratio and Percentage?

A ratio compares quantities, while a percentage expresses a quantity as a fraction of 100. For a ratio a:b, the percentages of the two parts out of their total are a/(a+b) × 100 and b/(a+b) × 100.

The ratio a:b represents a + b total parts. Therefore, the first part is a/(a+b) of the total and the second part is b/(a+b) of the total. For example, in the ratio 3:2, the total is 5 parts, so the parts are 3/5 × 100 = 60% and 2/5 × 100 = 40%.

Ratio and Percentage Formula & Tricks

Important Formulas

Ratio of two quantities to percentage
For a:b, first part % = a/(a+b) × 100; second part % = b/(a+b) × 100

Use the sum of the ratio terms as the total number of parts.

Percentage to ratio
p% = p:100 = p/100

Write the percentage over 100 and reduce the resulting ratio.

Ratio from two percentages
p%:q% = p:q

The common denominator 100 cancels when two percentages are compared.

Percentage of one quantity compared with another
Percentage = (quantity being compared / reference quantity) × 100

The denominator must be the reference or base quantity.

Remaining percentage
Remaining percentage = 100% − used percentage

This applies when the total quantity represents 100%.

Quick Tricks

Convert ratio parts directly into percentages

For a:b, add the terms first. Then multiply each term by 100 and divide by the sum.

Example: For 7:3, total parts = 10. The percentages are 7/10 × 100 = 70% and 3/10 × 100 = 30%.
Compare percentages by comparing their numbers

When both percentages have the same base, their ratio is simply the ratio of the numerical percentage values.

Example: 45%:30% = 45:30 = 3:2.
Use the complement for the remaining part

If one part is p% of the whole, the rest is (100 − p)%. Convert these two percentages into a ratio when required.

Example: If 35% of a quantity is spent, the spent-to-remaining ratio is 35:65 = 7:13.

Ratio and Percentage Concepts

Ratio of Parts as Percentage of the Total

If two quantities are in the ratio a:b, their percentages of the total are a/(a+b) × 100 and b/(a+b) × 100.

The ratio terms show parts, not percentages directly. Add the terms to obtain the total parts, then divide each term by this total. The resulting percentages always add to 100%.

Example: In the ratio 4:1, total parts = 5. The first quantity is 4/5 × 100 = 80%, and the second is 1/5 × 100 = 20%.

Converting a Percentage into a Ratio

To convert p% into a ratio, write it as p:100 and simplify.

The percentage p% means p/100. For a fractional percentage, first express it as a fraction or use an equivalent ratio before simplifying. If the question gives a percentage of a whole, the remaining percentage can be found using 100 − p.

Example: 62.5% = 62.5:100 = 625:1000 = 5:8.

Ratio Between Two Percentages

The ratio of p% and q% is p:q, provided both percentages refer to the same base.

Since p% = p/100 and q% = q/100, their common denominator cancels. This rule compares the percentage values themselves, not necessarily the original quantities if their bases are different.

Example: The ratio of 24% to 36% is 24:36 = 2:3.

Percentage Comparison of Quantities in a Ratio

For quantities in the ratio a:b, the first quantity is (a/b) × 100% of the second, while the second is (b/a) × 100% of the first.

This comparison uses one quantity as the reference base, so it differs from finding each quantity as a percentage of the total. For a ratio 3:2, the first is 150% of the second, and the second is 66⅔% of the first.

Example: If boys:girls = 3:2, boys are 3/2 × 100 = 150% of girls.

Ratio After a Percentage Change

If quantities in a ratio a:b change by x% and y%, their new ratio is a(100+x):b(100+y) for increases, with the signs adjusted for decreases.

Multiply a quantity increasing by x% by (100+x)/100. For a decrease of x%, multiply it by (100−x)/100. Common factors such as 1/100 can be removed before simplifying the new ratio.

Example: If A:B = 2:3, A increases by 20% and B decreases by 10%, the new ratio is 2 × 120 : 3 × 90 = 240:270 = 8:9.

Ratio and Percentage Video Lessons

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Duplicate, Triplicate and Sub-Duplicate Ratios

Learn the definitions and application of duplicate, triplicate, and sub-duplicate ratios, with clear methods for understanding how these special ratios are formed and used.

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Practice Ratio and Percentage Questions

Practise published questions related to this topic.

1If the profit on an item is 15% and the cost price is ₹800, find the selling price.→ 2A shopkeeper marks an item 40% above cost price and gives a discount of 25%. What is the profit percentage?→ 3Find the smallest positive integer that leaves a remainder of 2 when divided by 3, a remainder of 3 when divided by 4, and a remainder of 4 when divided by 5.→ 4After allowing a discount of 9% on the marked price of an article, it is sold for ₹53,053. Find its marked price (in ₹).→ 5The difference between cost price and selling price of an article is Rs. 520. Loss percentage is 20 percent. What is the cost price of the article?→ 6Two articles are sold at the same selling price. On one article, the trader makes a 25% profit, and on the other, he incurs a 25% loss. What is the overall percentage loss?→ 7An item is sold for ₹660 after allowing a discount of 20% on its marked price. Find the marked price of the item.→ 8Simplify: ((3⁵ × 6⁴ × 4³))/((3² × 6² × 4¹))→ 9What is the Highest Common Factor of 20, 30 and 50?→ 10A trader offers "Buy two, get one free" on a certain pair of trousers. If the marked price of the pair of trousers is ₹ 2700, and the trader still earns a profit of 20%, find the cost price of one pair of trousers for the trader.→

Ratio and Percentage Quick Quiz

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Quick Revision Notes

Ratio and Percentage Revision Points

Use these rules to convert ratios and percentages and to compare related quantities.

  • For a:b, total parts = a + b.
  • First part as a percentage of total = a/(a+b) × 100.
  • Second part as a percentage of total = b/(a+b) × 100.
  • p% = p:100; always simplify the ratio.
  • p%:q% = p:q when both percentages have the same base.
  • If one part is p% of the whole, the remaining part is (100−p)%.
  • For a:b, the first quantity is (a/b) × 100% of the second.
  • After percentage changes, multiply each ratio term by its corresponding change factor.

Ratio and Percentage FAQs

How do you convert the ratio 3:2 into percentages?

Add the terms: 3 + 2 = 5. The percentages are 3/5 × 100 = 60% and 2/5 × 100 = 40%.

How do you convert 37.5% into a ratio?

37.5% = 37.5:100 = 375:1000 = 3:8.

What is the ratio of 15% to 25%?

15%:25% = 15:25 = 3:5.

If A:B = 4:5, what percentage of the total is A?

Total parts = 4 + 5 = 9. Therefore, A is 4/9 × 100 = 44 4/9% of the total.

If 40% of a quantity is used, what is the used-to-remaining ratio?

The remaining percentage is 60%. Thus, used:remaining = 40:60 = 2:3.

If two quantities are in the ratio 5:4, what percentage is the first quantity of the second?

The first is 5/4 × 100 = 125% of the second.

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