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Ratio and Proportion - Concept of Ratio

Grade 7ICSE

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

🔑Concepts

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A ratio is a mathematical comparison of two quantities of the same kind, measured in the same units, obtained by dividing the first quantity by the second. Visually, it can be represented as a bar model where one bar's length is compared to another to see how many times larger or smaller it is.

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The two numbers in a ratio a:ba:b are called 'terms'. The first term aa is the 'Antecedent' and the second term bb is the 'Consequent'. If you visualize a group of 3 stars and 4 circles, the ratio of stars to circles is 3:43:4, where 3 is the antecedent.

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For a ratio to be meaningful, both quantities must be in the same units. For example, to compare 40 cm and 2 m, you must first convert 2 m to 200 cm so that the units cancel out, leaving a pure number ratio.

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A ratio is in its simplest form (or lowest terms) when the antecedent and consequent have no common factors other than 1. This is visually similar to simplifying a fraction where a large grid is grouped into the largest possible equal blocks to reduce the count.

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The order of terms in a ratio is extremely important. The ratio 2:52:5 is not the same as 5:25:2. Imagine a map scale where 1:1001:100 means 1 cm on the map equals 100 cm on the ground; reversing it to 100:1100:1 would change the meaning entirely.

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Since a ratio is a comparison of similar quantities, it is a 'dimensionless' quantity, meaning it has no units of its own. Whether comparing kilograms to kilograms or liters to liters, the final ratio is just a pair of numbers.

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Equivalent ratios are formed by multiplying or dividing both the antecedent and the consequent by the same non-zero number. This is like zooming in or out on a photograph; the height and width change, but the ratio between them stays the same.

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To divide a total quantity into a given ratio a:ba:b, the total is treated as consisting of a+ba + b equal parts. If you have a string and want to cut it in a 2:32:3 ratio, you would imagine the string divided into 5 equal segments, giving 2 segments to one part and 3 to the other.

📐Formulae

Ratio=First QuantitySecond QuantityRatio = \frac{\text{First Quantity}}{\text{Second Quantity}}

Ratio a to b=a:b=ab\text{Ratio } a \text{ to } b = a:b = \frac{a}{b}

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)}

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

Sum of Ratio Parts=a+b\text{Sum of Ratio Parts} = a + b

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

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

💡Examples

Problem 1:

Express the ratio of 75 paise to 3 rupees in its simplest form.

Solution:

Step 1: Convert both quantities to the same units. Since 1 rupee=100 paise1 \text{ rupee} = 100 \text{ paise}, then 3 rupees=3×100=300 paise3 \text{ rupees} = 3 \times 100 = 300 \text{ paise}. Step 2: Write the ratio as a fraction: 75300\frac{75}{300}. Step 3: Find the HCF of 75 and 300. The HCF is 75. Step 4: Divide both terms by 75: 75÷75300÷75=14\frac{75 \div 75}{300 \div 75} = \frac{1}{4}. Step 5: Write the result in ratio form: 1:41:4.

Explanation:

To compare different denominations, we must standardize the units first. Once units are identical, we simplify the resulting fraction by dividing by the highest common factor.

Problem 2:

Two numbers are in the ratio 5:85:8. If their sum is 390, find the two numbers.

Solution:

Step 1: Identify the ratio parts, which are 5 and 8. Step 2: Calculate the sum of the ratio parts: 5+8=135 + 8 = 13. Step 3: Calculate the value of one part: 39013=30\frac{390}{13} = 30. Step 4: Find the first number: 5×30=1505 \times 30 = 150. Step 5: Find the second number: 8×30=2408 \times 30 = 240. Verification: 150+240=390150 + 240 = 390.

Explanation:

The ratio 5:85:8 implies the total is divided into 13 equal units. By finding the value of a single unit, we can scale it back up to find the original numbers.