NCERT Solutions Class 6 Maths Ratio Proportion Ex 12.2

What is a Proportion? The Core Concept Explained

When we compare two ratios and find them to be completely equal, we say that they are in proportion. For instance, let us look at the ratios $10:20$ and $30:60$. If we simplify $10:20$ by dividing both terms by 10, we get the fraction $\frac{1}{2}$, which is $1:2$. Similarly, if we simplify $30:60$ by dividing both terms by 30, we also get $\frac{1}{2}$, which is $1:2$. Since both ratios simplify to the exact same value, they are equal. We represent this relation mathematically as:

$10:20 = 30:60$

In CBSE Class 6 Mathematics, we use a special double colon symbol ($::$) to represent proportion instead of an equal sign. We write this as:

$10:20 :: 30:60$

This is read aloud as '10 is to 20 as 30 is to 60'. When four numbers $a, b, c, d$ are in proportion, they have specific names based on their positions:

• Extreme Terms (or Extremes): The very first and fourth terms ($a$ and $d$).
• Middle Terms (or Means): The middle second and third terms ($b$ and $c$).

A golden rule of proportion to keep in mind is that the Product of Extremes must always equal the Product of Means ($a \times d = b \times c$).

Step-by-Step Method: How to Check for Proportions

1. Identify the Four Terms — Write down the four numbers in the exact order they are given in the problem. Let's label them as $a$, $b$, $c$, and $d$.
2. Form Two Separate Ratios — Group the first two numbers into the first ratio ($a:b$ or $\frac{a}{b}$) and the last two numbers into the second ratio ($c:d$ or $\frac{c}{d}$).
3. Simplify both Ratios — Reduce both fractions to their simplest terms by dividing the numerators and denominators by their Highest Common Factor (HCF).
4. Compare the Results — If the simplified forms are equal, the numbers are in proportion. Alternatively, verify if the Product of Extremes ($a \times d$) equals the Product of Means ($b \times c$).

CRITICAL EXAM TIP: Order of Terms Matters!

A common trap for students solving cbse class 6 ratio proportion ex 12 2 is altering the order of numbers given in the question.

If a question asks if $3, 10, 15, 50$ are in proportion, you must group them strictly in that sequence: $(3:10)$ and $(15:50)$.

• Ratio 1: $3:10$
• Ratio 2: $15:50 = 3:10$ (on dividing by 5)

Since $3:10 = 3:10$, they form a proportion.

If you rewrite them out of order, like $3, 15, 10, 50$, the ratios change and can lead to incorrect answers in word problems. Always keep the sequence exactly as specified in the problem statement!

Core Solved Examples from Exercise 12.2

• Example 1: Are 15, 45, 40 and 120 in proportion? Step 1: Write down the first ratio: $\frac{15}{45}$. Dividing both by 15, we get $\frac{1}{3}$ or $1:3$. Step 2: Write down the second ratio: $\frac{40}{120}$. Dividing both by 40, we get $\frac{1}{3}$ or $1:3$. Step 3: Since both ratios are equal, the numbers are in proportion. We write: $15:45 :: 40:120$.
• Example 2: Determine if 33, 44, 75, 100 are in proportion using the product method. Step 1: Identify terms: Extremes are $33$ and $100$. Means are $44$ and $75$. Step 2: Calculate Product of Extremes: $33 \times 100 = 3300$. Step 3: Calculate Product of Means: $44 \times 75 = 3300$. Step 4: Since Product of Extremes = Product of Means, the terms are in proportion.

Practice Questions with Solutions

• Q: Determine if the ratios 25 cm : 1 m and Rs 40 : Rs 160 are in proportion. A: Step 1: Convert units to make them identical. Since $1\text{ m} = 100\text{ cm}$, the first ratio is $25\text{ cm} : 100\text{ cm}$. Step 2: Simplify the first ratio: $\frac{25}{100} = \frac{1}{4} = 1:4$. Step 3: Simplify the second ratio: $\frac{40}{160} = \frac{4}{16} = \frac{1}{4} = 1:4$. Step 4: Compare both ratios. Since $1:4 = 1:4$, they form a proportion. Final answer: Yes, they are in proportion.
• Q: Check if the numbers 32, 48, 70, 210 are in proportion. A: Step 1: Form the first ratio: $\frac{32}{48}$. Dividing both by 16, we get $\frac{2}{3}$ or $2:3$. Step 2: Form the second ratio: $\frac{70}{210}$. Dividing both by 70, we get $\frac{1}{3}$ or $1:3$. Step 3: Compare both ratios. Since $2:3 \neq 1:3$, they are not equal. Final answer: No, they are not in proportion.
• Q: Write True (T) or False (F) for the statement: 5.2 : 3.9 :: 3 : 4. A: Step 1: Write the first decimal ratio as a fraction: $\frac{5.2}{3.9} = \frac{52}{39}$. Step 2: Simplify the fraction by dividing the numerator and denominator by 13: $\frac{52 \div 13}{39 \div 13} = \frac{4}{3}$ or $4:3$. Step 3: Compare the first ratio ($4:3$) with the second ratio ($3:4$). They are not equivalent. Final answer: False.
• Q: Find the value of x if 2 : 9 :: x : 27 is in proportion. A: Step 1: Apply the proportion property: Product of Extremes = Product of Means. Step 2: Extremes are $2$ and $27$. Means are $9$ and $x$. Step 3: Formulate the equation: $2 \times 27 = 9 \times x \implies 54 = 9x$. Step 4: Solve for $x$: $x = \frac{54}{9} = 6$. Final answer: x = 6.

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