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Ratio and Proportion

Practice MCQs

This chapter deals with solving problems of ratio & proportion. Ratio is a quantitative comparison of two or more quantities or values where as Proportion refers to the equality of two ratios.

Quantitative Aptitude Ratio and Proportion Competitive Exams

Ratio and Proportion is one of the most important topics in quantitative aptitude. It deals with comparison of quantities, equality of ratios, direct variation, inverse variation, and distribution of values in a given ratio. This topic is useful in questions on partnership, mixture, percentage, profit and loss, time and work, and data interpretation.

What are Ratio and Proportion?

A ratio is a comparison between two or more quantities of the same kind. If there are 12 boys and 8 girls in a class, the ratio of boys to girls is \(12:8\), which simplifies to \(3:2\).

A proportion is an equality of two ratios. If \(a:b = c:d\), then \(a\), \(b\), \(c\), and \(d\) are said to be in proportion.

Quick idea: Ratio compares quantities. Proportion says two ratios are equal.
Concept Meaning Example
Ratio Comparison of two quantities \(3:5\)
Proportion Equality of two ratios \(3:5 = 6:10\)
Antecedent First term of a ratio In \(a:b\), \(a\) is antecedent
Consequent Second term of a ratio In \(a:b\), \(b\) is consequent
Mean Proportional Middle proportional between two values \(\sqrt{ab}\)

“In ratio questions, always reduce the ratio to its simplest form before solving.”

Aptitude Tip
Key points
  • Ratio compares quantities of the same kind.
  • Ratio has no unit.
  • Proportion means equality of ratios.
  • In \(a:b\), \(a\) is antecedent and \(b\) is consequent.
  • In \(a:b = c:d\), \(ad = bc\).
  • Total parts method is very useful in distribution problems.
ratio proportion parts variation

Visual Understanding

These diagrams show how ratios compare quantities and how proportion connects equal ratios.

Ratio as Comparison
Boys : Girls = 3 : 2 3 parts 2 parts Ratio compares how many parts each quantity has
\[ 3:2 = \frac{3}{2} \]

A ratio shows how much one quantity is compared with another.

Proportion as Equal Ratios
3 : 5 First ratio = 6 : 10 Second ratio Both ratios reduce to the same value Therefore, they are in proportion
\[ 3:5 = 6:10 \]

If two ratios are equal, the four quantities are in proportion.

Dividing Amount in a Ratio
Divide ₹500 in the ratio 3 : 2 3 parts 2 Total parts = 3 + 2 = 5 One part = ₹500 ÷ 5 = ₹100
\[ \text{Share}=\frac{\text{Given part}}{\text{Total parts}}\times\text{Total amount} \]

First add the ratio parts, then calculate the value of one part.

Cross Multiplication in Proportion
a : b = c : d a b c d Product of extremes = Product of means
\[ a:b = c:d \Rightarrow ad = bc \]

Cross multiplication is the most direct method to solve proportion questions.

Important Formulas and Rules

Ratio Form
\[ a:b = \frac{a}{b} \]

Ratio can be written in fractional form.

Equivalent Ratio
\[ a:b = ka:kb \]

Multiplying both terms by the same number keeps the ratio same.

Proportion Rule
\[ a:b = c:d \Rightarrow ad=bc \]

Product of extremes equals product of means.

Fourth Proportional
\[ a:b = c:x \Rightarrow x=\frac{bc}{a} \]

Used to find the missing fourth term.

Third Proportional
\[ a:b = b:x \Rightarrow x=\frac{b^2}{a} \]

Used when the middle term is repeated.

Mean Proportional
\[ x=\sqrt{ab} \]

Mean proportional between \(a\) and \(b\).

Distribution Formula
\[ \text{Share}=\frac{\text{Part}}{\text{Total parts}}\times\text{Total} \]

Useful in dividing money, profit, or quantity.

Direct Variation
\[ x \propto y \]

If one quantity increases, the other also increases.

Inverse Variation
\[ x \propto \frac{1}{y} \]

If one quantity increases, the other decreases.

Duplicate Ratio
\[ a^2:b^2 \]

Duplicate ratio of \(a:b\).

Sub-Duplicate Ratio
\[ \sqrt{a}:\sqrt{b} \]

Sub-duplicate ratio of \(a:b\).

Compounded Ratio
\[ ac:bd \]

Compounded ratio of \(a:b\) and \(c:d\).

Rule: In proportion \(a:b=c:d\), \(a\) and \(d\) are extremes, while \(b\) and \(c\) are means. Therefore, \(ad=bc\).

Common Types of Questions

Simplifying Ratios

Convert quantities into the same unit and reduce the ratio.

  • Same unit comparison
  • HCF method
  • Fraction ratio
  • Decimal ratio
Dividing in a Ratio

Divide amount, profit, marks, or quantity using total parts.

  • Money sharing
  • Profit distribution
  • Quantity division
  • Age comparison
Finding Missing Term

Use cross multiplication to solve proportion.

  • Fourth proportional
  • Third proportional
  • Mean proportional
  • Unknown value
Variation Questions

Direct and inverse variation are common in aptitude exams.

  • Direct proportion
  • Inverse proportion
  • Work and time
  • Speed and time
Exam approach: Convert all quantities to the same unit before forming the ratio. Many mistakes happen because units are ignored.
Method Bank
Simplify Ratio

Divide both terms by their HCF.

\[ 24:36 = 2:3 \]
Divide ₹800 in \(3:5\)

Total parts \(=3+5=8\).

\[ \frac{3}{8}\times800=300,\quad \frac{5}{8}\times800=500 \]
Find Missing Term

Use cross multiplication.

\[ 3:5=12:x \Rightarrow 3x=60 \]
Mean Proportional

Between 4 and 9.

\[ \sqrt{4\times9}=6 \]

Tip: In ratio distribution, first find the value of one part. Then multiply by the required parts.

Ratio Solving Flow
Same Unit Simplify Use Parts Answer Convert units → simplify ratio → apply total parts
This flow works for most ratio distribution questions.
\[ \text{Total parts} = a+b+c+\cdots \]
\[ \text{One part}=\frac{\text{Total value}}{\text{Total parts}} \]

Step-by-Step Solving Method

Step Ratio Questions Proportion Questions
Step 1 Identify the quantities being compared. Write the two ratios clearly.
Step 2 Convert quantities to the same unit. Convert ratio into fraction form.
Step 3 Simplify using HCF. Apply cross multiplication.
Step 4 For distribution, add total parts. Solve for the unknown value.
Step 5 Find required share or comparison. Check the answer in the original proportion.
Important: A ratio is meaningful only when both quantities are expressed in the same unit.

Solved Examples

Question Method Answer
Simplify the ratio \(24:36\). Divide both terms by HCF \(12\).
\[ 24:36 = \frac{24}{12}:\frac{36}{12}=2:3 \]
2 : 3
Divide ₹600 between A and B in the ratio \(2:3\). Total parts:
\[ 2+3=5 \]
A's share:
\[ \frac{2}{5}\times600=240 \]
B's share:
\[ \frac{3}{5}\times600=360 \]
A = ₹240, B = ₹360
Find \(x\), if \(3:5 = 12:x\). Apply cross multiplication:
\[ 3x=5\times12 \]
\[ 3x=60 \Rightarrow x=20 \]
20
Find the fourth proportional to 4, 6, and 10. Let the fourth proportional be \(x\).
\[ 4:6 = 10:x \]
\[ 4x=60 \Rightarrow x=15 \]
15
Find the third proportional to 3 and 6. Let the third proportional be \(x\).
\[ 3:6 = 6:x \]
\[ 3x=36 \Rightarrow x=12 \]
12
Find the mean proportional between 4 and 25.
\[ x=\sqrt{4\times25}=\sqrt{100}=10 \]
10
The ratio of boys to girls is \(5:4\). If there are 45 students, find boys and girls. Total parts:
\[ 5+4=9 \]
One part:
\[ \frac{45}{9}=5 \]
Boys:
\[ 5\times5=25 \]
Girls:
\[ 4\times5=20 \]
Boys = 25, Girls = 20
If \(x:y = 2:3\) and \(y:z = 4:5\), find \(x:y:z\). Make \(y\) common. LCM of 3 and 4 is 12.
\[ x:y = 8:12 \]
\[ y:z = 12:15 \]
Therefore:
\[ x:y:z = 8:12:15 \]
8 : 12 : 15

Note: In combined ratio questions, make the common term equal before joining the ratios.

Common Traps and Shortcuts

Common Traps
  • Comparing quantities without converting them to the same unit.
  • Forgetting to simplify the ratio.
  • Adding ratio terms directly to the actual amount.
  • Confusing antecedent and consequent.
  • Applying direct proportion when the relation is inverse.
  • Not making the common term equal in combined ratios.
Useful Shortcuts
  • Always convert to same units first.
  • Use HCF to simplify ratios quickly.
  • In distribution, first find total parts.
  • For proportion, use cross multiplication.
  • For combined ratios, make the common term equal.
  • For mean proportional, use square root of product.
Exam approach: Ratio questions become easier when you treat each ratio value as a number of parts, not as the actual value.

Practice

A) Multiple Choice Questions
  1. Simplify the ratio \(18:24\).
    2 : 3 3 : 4 4 : 3 6 : 8
  2. Divide ₹900 in the ratio \(4:5\). What is the larger share?
    ₹300 ₹400 ₹500 ₹600
  3. Find \(x\), if \(2:3 = 8:x\).
    10 12 14 16
  4. Find the mean proportional between 9 and 16.
    10 11 12 13
  5. If \(a:b = 5:7\), then duplicate ratio is:
    10 : 14 25 : 49 7 : 5 49 : 25
B) Solve the Higher-Order Problems
  1. Divide ₹1200 among A, B, and C in the ratio \(2:3:5\). (Hint: Total parts \(=2+3+5\).)
  2. Find the fourth proportional to 6, 9, and 12. (Hint: \(6:9=12:x\).)
  3. Find the third proportional to 4 and 8. (Hint: \(4:8=8:x\).)
  4. If \(x:y=3:4\) and \(y:z=5:6\), find \(x:y:z\). (Hint: Make \(y\) common.)
  5. The ratio of income to expenditure is \(5:4\). If income is ₹25,000, find expenditure and savings. (Hint: One part \(=25000/5\).)
C) Match the Concept with the Correct Meaning
Concept Correct Meaning
Ratio Comparison of quantities
Proportion Equality of two ratios
Antecedent First term of a ratio
Consequent Second term of a ratio
Mean Proportional Square root of product of two numbers
Direct Proportion Both quantities increase or decrease together
Aptitude Reminder

Ratio and proportion questions are easiest when you use the parts method. For proportion, cross multiplication is the most reliable method.

Task: Create five questions using ratio simplification, amount division, fourth proportional, mean proportional, and combined ratio.

Show Suggested Answers
Multiple Choice
  1. 3 : 4
    Divide both terms by HCF \(6\).
    \[ 18:24 = 3:4 \]
  2. ₹500
    Total parts:
    \[ 4+5=9 \]
    Larger share:
    \[ \frac{5}{9}\times900=500 \]
  3. 12
    \[ 2:3 = 8:x \]
    \[ 2x=24 \Rightarrow x=12 \]
  4. 12
    \[ \sqrt{9\times16}=\sqrt{144}=12 \]
  5. 25 : 49
    Duplicate ratio of \(5:7\) is:
    \[ 5^2:7^2=25:49 \]
Higher-Order Problems
  1. Ratio \(2:3:5\), total amount ₹1200.
    \[ 2+3+5=10 \]
    One part:
    \[ \frac{1200}{10}=120 \]
    Shares:
    \[ A=240,\quad B=360,\quad C=600 \]
    Answer = A = ₹240, B = ₹360, C = ₹600.
  2. Fourth proportional to 6, 9, and 12:
    \[ 6:9=12:x \]
    \[ 6x=108 \Rightarrow x=18 \]
    Answer = 18.
  3. Third proportional to 4 and 8:
    \[ 4:8=8:x \]
    \[ 4x=64 \Rightarrow x=16 \]
    Answer = 16.
  4. Given:
    \[ x:y=3:4,\quad y:z=5:6 \]
    Make \(y\) common. LCM of 4 and 5 is 20.
    \[ x:y=15:20 \]
    \[ y:z=20:24 \]
    Answer = \(x:y:z=15:20:24\).
  5. Income : Expenditure \(=5:4\). Income = ₹25,000.
    \[ \text{One part}=\frac{25000}{5}=5000 \]
    Expenditure:
    \[ 4\times5000=20000 \]
    Savings:
    \[ 25000-20000=5000 \]
    Answer = Expenditure = ₹20,000, Savings = ₹5,000.
Concept Matching
  1. Ratio → Comparison of quantities
  2. Proportion → Equality of two ratios
  3. Antecedent → First term of a ratio
  4. Consequent → Second term of a ratio
  5. Mean Proportional → Square root of product of two numbers
  6. Direct Proportion → Both quantities increase or decrease together
Clue Explanation

Ratio compares quantities, while proportion compares two ratios. For sharing problems, use total parts. For missing value problems, use cross multiplication.

Exam tips
  • Convert all quantities to the same unit.
  • Simplify ratios using HCF.
  • In distribution, first add total parts.
  • For proportion, use cross multiplication.
  • For combined ratios, make the common term equal.
  • For mean proportional, use \(\sqrt{ab}\).
Practice MCQs