Ratios And Proportions — for Class 6-7

What Are Ratios — And Why Do They Matter?

A ratio compares two quantities of the same kind. When your school says “the pass-to-fail ratio is 4:1”, it means for every 4 students who passed, 1 failed. That’s it. No mystery.

We write ratios as $a:b$ (read: ” $a$ is to $b$ ”) or as a fraction $\frac{a}{b}$ . The key word is same kind — we compare runs scored to runs scored, not runs to wickets. Mixing units is the single most common mistake in Class 6-7 boards.

Ratios show up everywhere in real life: recipe ingredients, cricket statistics, map scales, mixing paints. Once you see how ratios work, you’ll start spotting them everywhere. And in board exams, this chapter carries reliable marks — easy to score if you know the pattern, easy to lose if you get careless.

Key Terms and Definitions

Ratio — a comparison of two quantities $a$ and $b$ (same unit), written as $a:b$ or $\frac{a}{b}$ .

Terms of a ratio — In $a:b$ , the number $a$ is the antecedent (first term) and $b$ is the consequent (second term).

Equivalent ratios — Ratios obtained by multiplying or dividing both terms by the same non-zero number. Just like equivalent fractions. $2:3 = 4:6 = 10:15$ .

Ratio in simplest form — When HCF of the two terms is 1. To simplify, divide both terms by their HCF.

Proportion — When two ratios are equal, we say they are in proportion. Written as $a:b :: c:d$ (read: ” $a$ is to $b$ as $c$ is to $d$ ”).

Extremes and Means — In $a:b :: c:d$ , the numbers $a$ and $d$ are extremes, and $b$ and $c$ are means.

Fourth Proportional — If $a:b :: c:x$ , then $x$ is the fourth proportional to $a$ , $b$ , $c$ .

Mean Proportional — If $a:x :: x:b$ , then $x = \sqrt{ab}$ is the mean proportional between $a$ and $b$ .

Always check units before writing a ratio. If Riya’s height is 150 cm and her bag is 5 kg, you cannot write a ratio between them. Ratios only make sense when both quantities are in the same unit.

Core Concepts and Methods

Simplifying Ratios

To simplify $a:b$ , find HCF $(a, b)$ and divide both terms.

Example: Simplify $36:48$ .

HCF $(36, 48) = 12$ .

36:48 = \frac{36}{12} : \frac{48}{12} = 3:4

When quantities have different units (but same kind), convert first, then ratio.

Example: Express the ratio of 75 paise to ₹3.

Convert ₹3 to paise: ₹3 = 300 paise.

75:300 = 1:4

Comparing Ratios

To compare $a:b$ and $c:d$ , convert both to fractions and compare.

Example: Which is greater — $3:4$ or $5:7$ ?

\frac{3}{4} = 0.75 \qquad \frac{5}{7} \approx 0.714

So $3:4 > 5:7$ .

Alternatively, cross-multiply: $3 \times 7 = 21$ vs $5 \times 4 = 20$ . Since $21 > 20$ , we get $3:4 > 5:7$ .

Cross multiplication is faster for comparing two ratios in exams. Just compare $ad$ vs $bc$ for $a:b$ and $c:d$ .

Dividing a Quantity in a Given Ratio

This is a very high-scoring question type in Class 6-7 boards.

Method: If we divide a quantity $Q$ in the ratio $a:b$ :

First part $= \frac{a}{a+b} \times Q$
Second part $= \frac{b}{a+b} \times Q$

Example: Divide ₹720 between Arjun and Meena in the ratio $5:4$ .

Total parts $= 5 + 4 = 9$ .

$\text{Arjun} = \frac{5}{9} \times 720 = \text{₹}400$ $\text{Meena} = \frac{4}{9} \times 720 = \text{₹}320$

Check: $400 + 320 = 720$ . ✓

The Cross-Product Rule for Proportions

If $a:b :: c:d$ , then:

a \times d = b \times c

Product of extremes = Product of means

This rule is the backbone of almost every proportion problem. If three of the four values are given, you find the fourth by cross-multiplying.

Example: Find $x$ if $4:7 :: x:35$ .

4 \times 35 = 7 \times x

140 = 7x

x = 20

Checking If Four Numbers Are in Proportion

Given numbers $a$ , $b$ , $c$ , $d$ — check if $a:b :: c:d$ .

Simply verify: is $a \times d = b \times c$ ?

Example: Are 6, 10, 9, 15 in proportion?

6 \times 15 = 90 \qquad 10 \times 9 = 90

Yes, they are in proportion.

Solved Examples

Easy (CBSE Class 6 Level)

Q1. In a class of 40 students, 24 are girls. Find the ratio of girls to boys.

Boys $= 40 - 24 = 16$ .

\text{Girls : Boys} = 24:16 = 3:2

Q2. Are 15, 25, 12, 20 in proportion?

Product of extremes $= 15 \times 20 = 300$ . Product of means $= 25 \times 12 = 300$ .

Since both products are equal, yes — they are in proportion.

Medium (CBSE Class 7 Level)

Q3. The ratio of zinc to copper in an alloy is $3:7$ . If the alloy weighs 500 g, find the weight of each metal.

Total parts $= 3 + 7 = 10$ .

\text{Zinc} = \frac{3}{10} \times 500 = 150 \text{ g}

\text{Copper} = \frac{7}{10} \times 500 = 350 \text{ g}

Q4. A map is drawn to a scale of $1:50000$ . What actual distance does 4 cm on the map represent?

1 \text{ cm on map} = 50000 \text{ cm in reality}

4 \text{ cm on map} = 4 \times 50000 = 200000 \text{ cm} = 2 \text{ km}

Map scale problems always use proportion. Identify which term represents map distance and which represents actual distance.

Q5. If $a:b = 2:3$ and $b:c = 4:5$ , find $a:c$ .

We need $b$ to be the same in both. Currently $b = 3$ in first ratio and $b = 4$ in second. Make $b$ equal by taking LCM $(3,4) = 12$ .

a:b = 8:12 \quad \text{(multiply by 4)}

b:c = 12:15 \quad \text{(multiply by 3)}

So $a:b:c = 8:12:15$ , giving $a:c = 8:15$ .

Hard (Class 7 / Mental Ability Level)

Q6. Two numbers are in the ratio $5:8$ . If 6 is subtracted from each, the ratio becomes $1:2$ . Find the numbers.

Let the numbers be $5k$ and $8k$ .

\frac{5k - 6}{8k - 6} = \frac{1}{2}

2(5k - 6) = 1(8k - 6)

10k - 12 = 8k - 6

2k = 6 \implies k = 3

Numbers are $5 \times 3 = 15$ and $8 \times 3 = 24$ .

Check: $(15-6):(24-6) = 9:18 = 1:2$ . ✓

Exam-Specific Tips

CBSE Class 6-7 Board Pattern: Ratios and Proportions typically appears in 1-mark (simplify ratio), 2-mark (divide quantity), and 3-mark (word problem) formats. The 3-mark problem is almost always a “divide in ratio” or “find the fourth proportional” question. Easy marks — don’t lose them to calculation errors.

Olympiad/Mental Ability (IMO, NTSE Stage 1): Expect compound ratio questions like $a:b:c$ problems, and “if ratio changes when a number is added/subtracted” — the harder format shown in Q6 above. Practice these separately.

For CBSE full marks in proportion questions:

Write the proportion clearly as $a:b :: c:d$
State “Product of extremes = Product of means”
Show all algebraic steps
Add a verification line at the end

This verification step costs you 10 seconds and secures the accuracy mark.

Common Mistakes to Avoid

Mistake 1: Comparing quantities with different units without converting. Writing “ratio of 2 km to 500 m = 2:500” is wrong. Convert both to metres first: 2000:500 = 4:1.

Mistake 2: Reversing antecedent and consequent. “Ratio of boys to girls” is NOT the same as “ratio of girls to boys.” Read the question word carefully. The order matters.

Mistake 3: Adding a constant to ratio directly. If the ratio is $3:5$ and you’re told to add 10 to each part, students write $3+10 : 5+10 = 13:15$ . But if the original numbers are $3k$ and $5k$ , you add 10 to $3k$ and $5k$ , not to $3$ and $5$ .

Mistake 4: Thinking ratio has units. Ratio is a pure number — no units. After writing $\frac{150 \text{ cm}}{200 \text{ cm}} = \frac{3}{4}$ , the answer is just $3:4$ , not ” $3:4$ cm”.

Mistake 5: Forgetting to verify proportions. Always do a quick cross-product check at the end. This catches arithmetic mistakes and gives you confidence — especially in 3-mark problems where you can’t afford to lose steps.

Practice Questions

Q1. Simplify the ratio: $\frac{2}{3} : \frac{4}{9}$

Convert to same denominator or multiply both by LCM(3,9) = 9: $\frac{2}{3} \times 9 = 6$ and $\frac{4}{9} \times 9 = 4$ . Answer: 3:2

Q2. Express 45 minutes to 2 hours as a ratio in simplest form.

Convert 2 hours = 120 minutes. Ratio = 45:120. HCF(45,120) = 15. 45÷15 = 3, 120÷15 = 8. Answer: 3:8

Q3. Find the mean proportional between 4 and 25.

Mean proportional $x = \sqrt{4 \times 25} = \sqrt{100} = 10$ . Answer: 10 Verify: $4:10 :: 10:25$ → $4 \times 25 = 100$ and $10 \times 10 = 100$ . ✓

Q4. Divide ₹1350 among Priya, Qusai, and Rahul in the ratio $2:3:4$ .

Total parts = 2 + 3 + 4 = 9. Priya = $\frac{2}{9} \times 1350$ = ₹300 Qusai = $\frac{3}{9} \times 1350$ = ₹450 Rahul = $\frac{4}{9} \times 1350$ = ₹600 Answer: ₹300 : ₹450 : ₹600

Q5. If $x:12 :: 9:4$ , find $x$ .

Product of extremes = Product of means: $x \times 4 = 12 \times 9$ $4x = 108$ $x = 27$ Answer: x = 27

Q6. The ages of Siya and Tara are in ratio $3:5$ . Three years later, the ratio becomes $2:3$ . Find their present ages.

Let present ages be $3k$ and $5k$ . After 3 years: $(3k+3):(5k+3) = 2:3$ $3(3k+3) = 2(5k+3)$ $9k + 9 = 10k + 6$ $k = 3$ Siya = 9 years, Tara = 15 years. Answer: Siya is 9 years old, Tara is 15 years old.

Q7. A recipe needs flour and sugar in ratio $7:2$ . How much flour is needed if you use 150 g of sugar?

$\frac{\text{flour}}{\text{sugar}} = \frac{7}{2}$ $\frac{\text{flour}}{150} = \frac{7}{2}$ flour $= \frac{7 \times 150}{2} = 525$ g Answer: 525 g of flour

Q8. Are the ratios $3:7$ and $15:35$ equivalent?

$\frac{15}{35}$ — divide both by 5 — gives $\frac{3}{7}$ . Yes, $3:7 = 15:35$ . They are equivalent ratios. Alternatively: $3 \times 35 = 105$ and $7 \times 15 = 105$ . Equal, so they are in proportion.

Frequently Asked Questions

What is the difference between a ratio and a fraction?

A fraction $\frac{a}{b}$ represents a part of a whole. A ratio $a:b$ compares two separate quantities. Numerically they’re written the same way, but conceptually a ratio can compare two independent things (boys vs girls in a class), while a fraction always relates a part to a whole.

Can a ratio be greater than 1?

Absolutely. If there are 7 boys and 3 girls, the ratio of boys to girls is $7:3$ , which as a fraction is $\frac{7}{3} > 1$ . Ratios have no restriction on size.

Why do we say “ratio has no units”?

When we compute $\frac{60 \text{ km}}{40 \text{ km}}$ , the “km” cancels out and we get $\frac{3}{2}$ , a pure number. This is why we must convert to the same unit before finding a ratio — so the units cancel cleanly.

What is a continued ratio?

When three or more quantities are compared together, like $a:b:c$ , that’s a continued ratio. We computed one in the worked example above ( $a:b:c = 8:12:15$ ). It’s not a separate concept — just an extended version.

How is proportion different from ratio?

A ratio is a comparison ( $2:3$ ). A proportion is a statement that two ratios are equal ( $2:3 :: 4:6$ ). Every proportion involves two ratios; not every ratio is part of a proportion.

My two ratios have a common middle term. How do I combine them?

Make the middle term equal (using LCM), then read off the combined ratio. This technique (shown in Q5 of Solved Examples) comes up frequently in Class 7 and is a must-know for Olympiad prep.

How do I know when to use proportion vs simple multiplication?

When the relationship between two quantities is constant (more workers → more work, more speed → less time), set up a proportion. For direct proportion, both quantities increase together. For inverse proportion, one increases as the other decreases — you’ll study inverse proportion formally in Class 8.

What if the ratio involves decimals or fractions?

Clear the decimals or fractions first by multiplying both terms by the appropriate power of 10 or the LCM of denominators. Then simplify as usual. Example: $0.5:1.5 = 5:15 = 1:3$ .