Grade 6 Ratio - Proportion

Step 1: Write the ratio as a fraction: \( \frac{12}{18} \).

Step 2: Find the greatest common factor (GCF) of 12 and 18.

Prime factors of 12: \( 12 = 2^2 \times 3^1 \).

Prime factors of 18: \( 18 = 2^1 \times 3^2 \).

Common factors: \( 2^1 \times 3^1 \), so GCF = \( 2 \times 3 = 6 \).

Step 3: Divide both terms by the GCF:

\( \frac{12 \div 6}{18 \div 6} = \frac{2}{3} \).

Step 4: Write the simplified ratio: \( 2:3 \).

Step 5: Therefore, the simplified ratio is \( 2:3 \).

Step 1: Write the proportion: \( \frac{3}{4} = \frac{x}{12} \).

Step 2: Cross-multiply to eliminate fractions:

\( 3 \times 12 = 4 \times x \).

\( 36 = 4x \).

Step 3: Solve for \( x \) by dividing both sides by 4:

\( \frac{36}{4} = \frac{4x}{4} \).

\( x = 9 \).

Step 4: Verify by substituting \( x = 9 \) back into the proportion:

\( \frac{3}{4} = \frac{9}{12} \).

Simplify \( \frac{9}{12} = \frac{3}{4} \), which is true.

Step 5: Therefore, the solution is \( x = 9 \).

Step 1: Set up the proportion comparing weight to cost:

\( \frac{2 \text{ kg}}{₹120} = \frac{5 \text{ kg}}{x \text{ ₹}} \).

Step 2: Cross-multiply:

\( 2 \times x = 120 \times 5 \).

\( 2x = 600 \).

Step 3: Solve for \( x \):

\( x = \frac{600}{2} = 300 \).

Step 4: Verify:

Unit cost: \( \frac{120}{2} = 60 \) ₹/kg.

For 5 kg: \( 60 \times 5 = 300 \) ₹, which matches.

Step 5: Therefore, 5 kg of apples cost ₹300.

Step 1: Simplify the first ratio, 4:6.

Write as a fraction: \( \frac{4}{6} \).

GCF of 4 and 6 is 2.

\( \frac{4 \div 2}{6 \div 2} = \frac{2}{3} \), so 4:6 = 2:3.

Step 2: Simplify the second ratio, 8:12.

Write as a fraction: \( \frac{8}{12} \).

GCF of 8 and 12 is 4.

\( \frac{8 \div 4}{12 \div 4} = \frac{2}{3} \), so 8:12 = 2:3.

Step 3: Compare the simplified ratios.

Both ratios simplify to 2:3, so they are equivalent.

Step 4: Alternative method: Check cross-products.

For 4:6 = 8:12, cross-multiply: \( 4 \times 12 = 6 \times 8 \).

\( 48 = 48 \), which is true.

Step 5: Therefore, the ratios 4:6 and 8:12 are equivalent.

Step 1: Set up the proportion comparing distance to time:

\( \frac{120 \text{ km}}{2 \text{ hours}} = \frac{x \text{ km}}{5 \text{ hours}} \).

Step 2: Cross-multiply:

\( 120 \times 5 = 2 \times x \).

\( 600 = 2x \).

Step 3: Solve for \( x \):

\( x = \frac{600}{2} = 300 \).

Step 4: Verify:

Speed: \( \frac{120}{2} = 60 \) km/h.

For 5 hours: \( 60 \times 5 = 300 \) km, which matches.

Step 5: Therefore, the car will travel 300 km in 5 hours.