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Proportional Reasoning-1 - Ratios in their Simplest Form

Grade 8CBSE

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

🔑Concepts

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A ratio is a comparison of two quantities of the same kind by division. It is denoted as a:ba : b or ab\frac{a}{b}.

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

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A ratio must always be expressed between quantities of the same units. For example, to find the ratio between 2 kg2 \text{ kg} and 500 g500 \text{ g}, both must be converted to grams or kilograms.

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A ratio a:ba : b is in its simplest form if the Highest Common Factor (HCF) of aa and bb is 11.

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Multiplying or dividing both terms of a ratio by the same non-zero number produces an equivalent ratio.

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A ratio does not have any units; it is a dimensionless number.

📐Formulae

Ratio=Quantity 1Quantity 2\text{Ratio} = \frac{\text{Quantity 1}}{\text{Quantity 2}}

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

Sum of parts in ratio a:b=a+b\text{Sum of parts in ratio } a:b = a + b

Value of part a=aa+b×Total Quantity\text{Value of part } a = \frac{a}{a+b} \times \text{Total Quantity}

💡Examples

Problem 1:

Express the ratio 144:180144 : 180 in its simplest form.

Solution:

Find the HCF of 144144 and 180180. The factors of 144144 and 180180 give an HCF=36\text{HCF} = 36. Dividing both terms by 3636: 144÷36180÷36=45\frac{144 \div 36}{180 \div 36} = \frac{4}{5} Therefore, the simplest form is 4:54 : 5.

Explanation:

To simplify a ratio, divide both the antecedent and the consequent by their Highest Common Factor.

Problem 2:

Find the ratio of 75 paise75 \text{ paise} to ₹3₹ 3 in simplest form.

Solution:

First, convert both to the same units. ₹1=100 paise₹ 1 = 100 \text{ paise}, so ₹3=300 paise₹ 3 = 300 \text{ paise}. The ratio is 75:30075 : 300. Ratio=75300=75÷75300÷75=14\text{Ratio} = \frac{75}{300} = \frac{75 \div 75}{300 \div 75} = \frac{1}{4} The simplest form is 1:41 : 4.

Explanation:

Ratios can only be computed when the units are identical. Here, rupees are converted to paise before division.

Problem 3:

Divide ₹2500₹ 2500 between Alice and Bob in the ratio 3:23 : 2.

Solution:

Sum of the ratio parts =3+2=5= 3 + 2 = 5. Alice's share =35×2500=3×500=1500= \frac{3}{5} \times 2500 = 3 \times 500 = 1500. Bob's share =25×2500=2×500=1000= \frac{2}{5} \times 2500 = 2 \times 500 = 1000. To verify: 1500+10002500\begin{array}{r} 1500 \\ + 1000 \\ \hline 2500 \end{array} Alice gets ₹1500₹ 1500 and Bob gets ₹1000₹ 1000.

Explanation:

To divide a quantity in a given ratio, first find the total number of parts, then find the value of each part relative to the total.