Ratio and Proportion

🔹 What is a Ratio?

A ratio is a way to compare two quantities of the same kind. It shows how many times one quantity contains another. Ratios help in comparing values and understanding their relative sizes.

Ratio Notations:

• Colon form: 2:3

• Fraction form: 2/3

• Word form: 2 to 3

📌 Example:
If a recipe uses 2 cups of flour and 3 cups of sugar, the ratio of flour to sugar is 2:3 or 2/3.

🔹 Types of Ratios

1. Simple Ratio:
   Compares two values directly.
   👉 Example: 4:5 or 3:7

2. Compound Ratio:
   Compares more than two quantities.
   👉 Example: 2:3:4 (e.g., managers:team leads:employees)

3. Equivalent Ratio:
   Ratios that represent the same relationship.
   👉 Example: 2:3 = 4:6 = 6:9

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🔹 What is Proportion?

A proportion is a statement that two ratios are equal.

$$\frac{a}{b}=\frac{c}{\mathrm{d}}$$

It helps solve problems related to scaling, sharing, or comparison.

📌 Example:

$$\frac{2}{3}=\frac{4}{\mathrm{6}}$$

Cross-multiplying: 2 × 6 = 3 × 4 → 12 = 12 ✅ Proportion is valid.

🔹 Types of Proportions

1. Direct Proportion:
   When one value increases, the other also increases in the same ratio.
   👉 Example: If 5 pencils cost ₹2, then 10 pencils cost ₹4.

2. Inverse Proportion:
   When one value increases, the other decreases.
   👉 Example: If 4 workers take 6 hours to finish a job, 8 workers will take 3 hours.

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🔹 Examples

🔸 Ratio Example:

Compound Ratio:
A company staff ratio is 2:3:5 → For every 2 managers, there are 3 team leads and 5 employees.

🔸 Direct Proportion Example:

5 pencils = ₹2
How much for 10 pencils?
Set up proportion:

$$\frac{5}{2}=\frac{10}{x}\Rightarrow x=\mathrm{<b>4</b>}$$

✅ 10 pencils cost ₹4.

🔸 Inverse Proportion Example:

4 workers take 6 hours → How many for 3 hours?

$$4\times 6=x\times 3\Rightarrow x=\mathrm{<b>8</b>}$$

✅ 8 workers are needed.

📝 Practice Questions (MCQs)

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1. What is the ratio of 24 to 60 in simplest form?

A) 2:5
B) 3:8
C) 4:10
D) 6:15

✅ Answer: A) 2:5
Explanation:
GCD of 24 and 60 is 12
24 ÷ 12 = 2, 60 ÷ 12 = 5
Simplified ratio = 2:5

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2. If 3:4 = x:12, what is the value of x?

A) 6
B) 7
C) 8
D) 9

✅ Answer: C) 8
Explanation:

$$\frac{3}{4}=\frac{x}{12}\Rightarrow 3\times 12=4\times x\Rightarrow 36=4x\Rightarrow x=9$$

Oops! Let's recalculate:
Actually,

$$\frac{3}{4}=\frac{x}{12}\Rightarrow x=(3\times 12)\mathrm{/}4=36\mathrm{/}4=<b>9</b>$$

✅ Correct Answer: D) 9

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3. The cost of 8 pens is ₹56. What is the cost of 14 pens (in direct proportion)?

A) ₹92
B) ₹98
C) ₹100
D) ₹104

✅ Answer: B) ₹98
Explanation:

$$\frac{8}{56}=\frac{14}{x}\Rightarrow 8x=56\times 14=784\Rightarrow x=784\mathrm{/}8=<b>98</b>$$

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4. If 6 men can complete a job in 12 days, how many men are needed to do the same job in 8 days?

A) 8
B) 9
C) 10
D) 12

✅ Answer: A) 9
Explanation (Inverse Proportion):
More men → fewer days:
6 × 12 = x × 8 → x = (6 × 12) / 8 = 72 / 8 = 9 men

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5. Which of the following is an equivalent ratio to 3:5?

A) 6:10
B) 9:12
C) 12:15
D) Both A and C

✅ Answer: D) Both A and C
Explanation:
3:5 = 6:10 (×2), 3:5 = 12:15 (×4) → Both are equivalent.