krit.club logo

Proportional Reasoning-1 - Ratios

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

A ratio is a comparison of two quantities of the same kind and in the same units, obtained by dividing one quantity by the other.

•

For two quantities aa and bb, the ratio is written as a:ba:b (read as 'aa is to bb') or as a fraction ab\frac{a}{b}.

•

In the ratio a:ba:b, the first term aa is called the antecedent and the second term bb is called the consequent.

•

A ratio has no units. It is a pure number comparison.

•

Ratios can be simplified by dividing both the antecedent and the consequent by their Highest Common Factor (HCF). A ratio is in the simplest form if HCF(a,b)=1HCF(a, b) = 1.

•

To compare two ratios, convert them into fractions and then into like fractions (fractions with the same denominator).

•

Equivalent ratios are obtained by multiplying or dividing the antecedent and consequent by the same non-zero number.

📐Formulae

Ratio=Quantity AQuantity B\text{Ratio} = \frac{\text{Quantity A}}{\text{Quantity B}}

Simplest Form of a:b=a÷HCF(a,b)b÷HCF(a,b)\text{Simplest Form of } a:b = \frac{a \div \text{HCF}(a,b)}{b \div \text{HCF}(a,b)}

Part 1=aa+b×Total Quantity\text{Part 1} = \frac{a}{a+b} \times \text{Total Quantity}

Part 2=ba+b×Total Quantity\text{Part 2} = \frac{b}{a+b} \times \text{Total Quantity}

Equivalent Ratio: a:b=(a×k):(b×k) where k≠0\text{Equivalent Ratio: } a:b = (a \times k) : (b \times k) \text{ where } k \neq 0

💡Examples

Problem 1:

Express the ratio of 80 cm80 \text{ cm} to 2.4 m2.4 \text{ m} in simplest form.

Solution:

First, convert both quantities to the same unit. Since 1 m=100 cm1 \text{ m} = 100 \text{ cm}, 2.4 m=2.4×100=240 cm2.4 \text{ m} = 2.4 \times 100 = 240 \text{ cm} Now, find the ratio: Ratio=80240=80÷80240÷80=13\text{Ratio} = \frac{80}{240} = \frac{80 \div 80}{240 \div 80} = \frac{1}{3} The ratio is 1:31:3.

Explanation:

Ratios must always be calculated using the same units. Here, we converted meters to centimeters before dividing.

Problem 2:

Divide ₹5600₹ 5600 between Rita and Sita in the ratio 3:43:4.

Solution:

Sum of the ratio parts =3+4=7= 3 + 4 = 7. Total amount =₹5600= ₹ 5600. Rita's share: 37×5600=3×800=2400\frac{3}{7} \times 5600 = 3 \times 800 = 2400 Sita's share: 47×5600=4×800=3200\frac{4}{7} \times 5600 = 4 \times 800 = 3200 Rita gets ₹2400₹ 2400 and Sita gets ₹3200₹ 3200.

Explanation:

To divide a quantity into a ratio a:ba:b, we calculate the fraction of the total for each part using the sum of the terms.

Problem 3:

Which ratio is greater: 5:85:8 or 7:127:12?

Solution:

Convert the ratios into fractions: 58 and 712\frac{5}{8} \text{ and } \frac{7}{12} Find the LCM of the denominators 88 and 1212, which is 2424. Convert to like fractions: 5×38×3=1524\frac{5 \times 3}{8 \times 3} = \frac{15}{24} 7×212×2=1424\frac{7 \times 2}{12 \times 2} = \frac{14}{24} Since 15>1415 > 14, therefore 1524>1424  ⟹  5:8>7:12\frac{15}{24} > \frac{14}{24} \implies 5:8 > 7:12

Explanation:

Comparison of ratios is done by making the denominators the same, just like comparing fractions.

Problem 4:

In a class, the ratio of boys to girls is 5:35:3. If there are 2424 girls, find the number of boys.

Solution:

Let the number of boys be 5x5x and the number of girls be 3x3x. Given: 3x=243x = 24 x=243=8x = \frac{24}{3} = 8 Number of boys =5x=5×8=40= 5x = 5 \times 8 = 40.

Explanation:

By using a common multiplier xx, we can represent quantities in a ratio as algebraic terms to solve for unknown values.