Ratio: Basics, Formulas, Rules and Solved Questions

Ratio compares two quantities of the same kind by division. This page covers ratio notation, terms, equivalent ratios, simplification, comparison, division of quantities and compound ratios. It also includes direct formulas, useful shortcuts and numerical examples for solving common ratio questions accurately.

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What is Ratio?

A ratio is a comparison of two quantities of the same kind using division. It is written as a:b or a/b, where a is the first term and b is the second term, with b not equal to zero.

In the ratio a:b, a is called the antecedent and b is called the consequent. Both quantities must be expressed in the same units before forming the ratio. For example, 2 metres:50 centimetres becomes 200:50 = 4:1. A ratio has no unit after the units of both quantities are cancelled.

Ratio Formula & Tricks

Important Formulas

Basic ratio
a:b = a/b

The ratio of a to b is the quotient of a divided by b.

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

Multiplying or dividing both terms by the same non-zero number gives an equivalent ratio.

Simplest form of a ratio
a:b = (a ÷ d):(b ÷ d), where d = HCF(a,b)

Divide both terms by their highest common factor to reduce the ratio to its simplest form.

Division of a total in a ratio
First share = T × a/(a+b); Second share = T × b/(a+b)

If a total T is divided in the ratio a:b, the total number of parts is a+b.

Fourth proportional
If a:b = c:x, then x = bc/a

The fourth proportional x is found by cross-multiplication, provided a is non-zero.

Quick Tricks

Convert units before forming a ratio

A ratio can be formed directly only when both quantities have the same units. Convert the larger or smaller unit first, then simplify.

Example: 3 kg:750 g = 3000:750 = 4:1.
Use the difference of terms for ratio questions

If two quantities are in the ratio a:b, their difference represents b−a parts. One part equals the actual difference divided by b−a.

Example: If two numbers are in the ratio 5:8 and differ by 21, one part is 21/(8−5) = 7. The numbers are 35 and 56.
Scale ratios using a common multiplier

To match one term with a known value, multiply every term by the same factor. This avoids calculating the ratio again.

Example: If x:y = 3:7 and x = 24, the multiplier is 24/3 = 8, so y = 7×8 = 56.

Ratio Concepts

Terms and Simplest Form of a Ratio

The two quantities in a ratio are its terms: the first is the antecedent and the second is the consequent. A ratio is in simplest form when its terms have no common factor other than 1.

For a:b, a is the antecedent and b is the consequent. To simplify a ratio, divide both terms by their HCF. The order of terms must not be changed because a:b is generally different from b:a.

Example: 18:24 = (18÷6):(24÷6) = 3:4. However, 4:3 is not equivalent to 3:4.

Equivalent Ratios

Equivalent ratios have the same value and are obtained by multiplying or dividing both terms by the same non-zero number.

If a:b = c:d, then ad = bc. This cross-product rule checks whether two ratios are equivalent. For example, 4:6 and 10:15 are equivalent because 4×15 = 6×10 = 60.

Example: 7:9 = 28:36 because both terms of 7:9 are multiplied by 4.

Comparing Two Ratios

To compare a:b and c:d, compare the cross-products ad and bc, or convert both ratios to a common form.

If ad > bc, then a:b > c:d. If ad < bc, then a:b < c:d. This method avoids converting ratios into decimal values and works for positive quantities.

Example: Compare 5:8 and 3:5. Since 5×5 = 25 and 8×3 = 24, 5:8 is greater than 3:5.

Dividing a Quantity in a Given Ratio

To divide T in the ratio a:b, add the ratio terms to get a+b equal parts, then assign a parts and b parts to the two shares.

The first share is Ta/(a+b), and the second share is Tb/(a+b). For more than two terms, if T is divided in the ratio a:b:c, the shares are Ta/(a+b+c), Tb/(a+b+c) and Tc/(a+b+c).

Example: Divide ₹720 in the ratio 5:7. Total parts = 12; shares are 720×5/12 = ₹300 and 720×7/12 = ₹420.

Compound Ratio

The compound ratio of a:b and c:d is ac:bd, formed by multiplying the antecedents and the consequents separately.

For three ratios, a:b, c:d and e:f, the compound ratio is ace:bdf. The ratios should be expressed consistently before multiplication. The result may be simplified after forming the compound ratio.

Example: The compound ratio of 2:3 and 4:5 is (2×4):(3×5) = 8:15.

Ratio 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 Questions

Practise published questions related to this topic.

1The marked price of a cooker is the same at four shops I, II, III and IV. Shop I allows two successive discounts of 87% and 37%, shop II allows successive discounts of 81% and 37%, shop III allows successive discounts of 57% and 54% and shop IV allows successive discounts of 91%, 25%, and 59% on the marked price of the cooker. Which shop is selling the cooker at the lowest price?→ 2Two numbers are in the ratio of 2 : 5. If the Highest Common Factor of the numbers is 11, then the sum of the numbers will be _____.→ 3To gain 25% after allowing a discount of 10%, the shopkeeper must mark the price of the article which costs him ₹ 360 as -→ 4An article is marked for ₹2,000. A 25% discount is offered on the marked price, followed by a second discount of 10% on the new price. What is the final selling price of the article?→ 5An article was sold at ₹1260 with a 5% loss. What was the cost price?→ 6A shopkeeper fixes the marked price of an item 150% above its cost price. What percentage of discount on marked price should be allowed to gain 110%?→ 7The marked price of an article is ₹13,125. The shopkeeper gave a 20% discount and still makes a profit of 5%. What is the cost price (in ₹) of an article?→ 8The Highest Common Factor of two numbers is 12 and their Least Common Multiple is 180. The ratio of the numbers is 5 : 3. What is the sum of the two numbers?→ 9A dealer allowed a 23% discount on an article and was still able to manage a 44% profit. Find the ratio of the cost price to the listed price.→ 10A student bought a calculator for ₹400 and sold it for ₹440. What is the profit percentage?→

Ratio Quick Quiz

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Quick Revision Notes

Ratio Revision Points

Use these rules and formulas for quick revision of ratio questions.

  • A ratio compares quantities of the same kind and requires the same units.
  • In a:b, a is the antecedent and b is the consequent.
  • Simplify a ratio by dividing both terms by their HCF.
  • Multiplying or dividing both terms by the same non-zero number gives an equivalent ratio.
  • For a:b = c:d, the cross-products satisfy ad = bc.
  • To divide T in a:b, use Ta/(a+b) and Tb/(a+b).
  • To compare a:b and c:d, compare ad with bc.
  • The compound ratio of a:b and c:d is ac:bd.

Ratio FAQs

How do you simplify a ratio?

Divide both terms by their HCF. For example, 45:60 = 3:4 after dividing both terms by 15.

What is the ratio of 2 hours to 30 minutes?

Convert 2 hours to 120 minutes. The ratio is 120:30 = 4:1.

How do you find the fourth proportional to 6, 9 and 14?

If 6:9 = 14:x, then x = (9×14)/6 = 21.

How is ₹1,260 divided in the ratio 4:5?

Total parts = 9. The shares are 1260×4/9 = ₹560 and 1260×5/9 = ₹700.

How can two ratios be compared without converting them to decimals?

Use cross-products. For a:b and c:d, compare ad and bc. For 7:12 and 5:9, 7×9 = 63 and 12×5 = 60, so 7:12 is greater.

What is the difference between a ratio and an equivalent ratio?

A ratio is a comparison such as 3:5. An equivalent ratio has the same value, such as 6:10, obtained by multiplying both terms of 3:5 by 2.

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