Ratio, Proportion and Unitary Method Chapter Notes - Class 6 ICSE PDF Download

Table of Contents
1. Introduction
2. Ratio
3. Simplification of Ratios
4. Dividing a Quantity in a Given Ratio
5. Comparison of Ratios
6. Proportion
7. Unitary Method
8. Variation
9. Applications and tips
10. Summary
View more

Chapter Notes: Ratio, Proportion, and Unitary Method

Chapter Notes: Ratio, Proportion, and Unitary Method

Introduction

Comparing quantities is part of everyday life: we compare marks, heights, prices and many other things. The ideas of ratio, proportion and the unitary method give systematic, reliable ways to make such comparisons and to solve practical problems. This chapter explains these ideas step by step, with clear definitions, methods and worked examples so you can apply them to school problems and real-life situations.

Ratio

What is a ratio?

Definition: A ratio compares two quantities of the same kind by division. If the two quantities are x and y, the ratio is written as x:y and can also be written as the fraction $\dfrac{x}{y}$

Notation and terms

• Notation:x:y where x is the antecedent (first term) and y is the consequent (second term).
• Types: part-to-part (e.g., boys:girls), part-to-whole (e.g., boys:total), ratio of rates (e.g., speed1:speed2).
• Important: Ratios compare like quantities and so have no units.

How to form and read a ratio

• Ensure both quantities are of the same unit (convert if needed).
• Divide the first quantity by the second to get the numerical ratio.
• Write the result as a:b or as a fraction $\dfrac{a}{b}$.

Key properties of ratios

• Order matters: a:b is usually different from b:a.
• Equivalent ratios: multiplying or dividing both terms by the same non-zero number does not change the ratio (e.g., 2:3 = 4:6).
• Both terms must be non-zero to form a meaningful ratio.

Worked example - ratio with unit conversion

Example: Find the ratio between 2.75 kg and 750 g.

Sol.

Convert 2.75 kg to grams:
$2.75 \times 1000 = 2750\ \mathrm{g}$

Form the ratio:
$2750:750$ which as a fraction is $\dfrac{2750}{750}$.

Simplify the fraction by dividing numerator and denominator by their GCD (250):
$\dfrac{2750}{750} = \dfrac{11}{3}$.

Therefore, the ratio is $11:3$.

Difference between a fraction and a ratio

Both fractions and ratios involve division but they represent different ideas.

• A fraction such as $\dfrac{3}{5}$ denotes "3 parts out of 5 equal parts of a whole".
• A ratio such as 3:5 compares two separate quantities: for every 3 units of the first quantity there are 5 units of the second quantity.

Simplification of Ratios

Definition: To simplify a ratio is to express it in lowest terms by removing any common factors from the two terms.

Steps to simplify a ratio

• Write the ratio as a fraction $\dfrac{x}{y}$
• Divide both numerator and denominator by their greatest common divisor (GCD).
• If the terms are fractional expressions, first clear denominators by multiplying both terms by the LCM of denominators, then simplify.

Note: Multiplying or dividing both terms by the same non-zero number keeps the ratio equivalent.

Worked examples - simplification

Example 1: Simplify the ratio 35:75.

Sol.

Write as a fraction:
$\dfrac{35}{75}$.

Find GCD of 35 and 75, which is 5, and divide both terms by 5:
$\dfrac{35}{75} = \dfrac{7}{15}$.

Hence, the simplified ratio is $7:15$.

Example 2: Simplify the ratio $\dfrac{3}{20} : \dfrac{1}{40}$

Sol.

Clear denominators by multiplying both terms by LCM of 20 and 40, which is 40.

Calculate:
$\left(\dfrac{3}{20} \times 40\right) : \left(\dfrac{1}{40} \times 40\right)$

Evaluate:
$\left(3 \times 2\right) : 1 = 6:1$

So the simplified ratio is $6:1$

Dividing a Quantity in a Given Ratio

Definition: When a total amount is shared between two (or more) persons in a given ratio, each person's share is found by dividing the total in proportion to the parts of the ratio.

Formula (for two parts):

$\text{First part} = \dfrac{x}{x+y} \times a$

$\text{Second part} = \dfrac{y}{x+y} \times a$

Steps

• Add the terms of the ratio to get the total number of parts.
• Find the fraction of the total that each part represents (term ÷ total parts).
• Multiply each fraction by the total quantity to get each share.

Worked examples - division in a ratio

Example 1: A rope of length 150 cm is divided in the ratio 3:7. Find each piece's length.

Sol.

Sum of ratio parts is $3+7=10$.

First piece $= \dfrac{3}{10}\times 150 = 45\ \mathrm{cm}$.

Second piece $= \dfrac{7}{10}\times 150 = 105\ \mathrm{cm}$.

Example 2: Divide ₹1500 in the ratio 12:13.

Sol.

Sum of ratio parts is $12+13=25$.

First share $= \dfrac{12}{25}\times 1500 =$ ₹720.

Second share $= \dfrac{13}{25}\times 1500 =$ ₹780.

Comparison of Ratios

Definition: To compare two ratios, convert each to a fraction and compare the fractions by cross-multiplication.

Rule: For ratios $\dfrac{p}{q}$ and $\dfrac{r}{s}$, compare $p \times s$ and $q \times r$.

Steps:

• Convert ratios to fractions (e.g., a:b = a/b).
• Cross-multiply: For p/q and r/s, compare p × s and q × r.
• Determine the relationship based on the formula.

Worked examples - comparison

Example 1: Compare the ratios 5:9 and 6:11.

Sol.

Write as fractions $\dfrac{5}{9}$ and $\dfrac{6}{11}$.

Cross-multiply: $5 \times 11 = 55$ and $9 \times 6 = 54$.

Since $55 > 54$, we have $\dfrac{5}{9} > \dfrac{6}{11}$.

Thus 5:9 > 6:11.

Example 2: Compare 30:65 and 6:13.

Sol.

Write as fractions $\dfrac{30}{65}$ and $\dfrac{6}{13}$.

Cross-multiply: $30 \times 13 = 390$ and $65 \times 6 = 390$.

Since both products are equal, the ratios are equal: $30:65 = 6:13$.

Proportion

Definition: Two equal ratios form a proportion. If $p:q$ and $r:s$ are equal, we write $p:q :: r:s$ and read "$p$ is to $q$ as $r$ is to $s$".

Terms used in a proportion

• Extremes: first and fourth terms (p and s).
• Means: second and third terms (q and r).
• Property: product of extremes equals product of means: $p \times s = q \times r$

How to check or use a proportion

• Convert ratios to fractions and compare, or cross-multiply and check equality.
• Use the property of products to find an unknown term when three terms are given.
• Ensure corresponding terms are quantities of the same kind (e.g., both lengths or both amounts).

Worked examples - proportion

Example 1: Check if 15:24 and 10:16 form a proportion.

Sol.

Write as fractions $\dfrac{15}{24}$ and $\dfrac{10}{16}$.

Simplify both: $\dfrac{15}{24} = \dfrac{5}{8}$ and $\dfrac{10}{16} = \dfrac{5}{8}$.

Since both are equal, $15:24 :: 10:16$ and they form a proportion.

Example 2: Find $x$ if 7, 33, $x$ and 66 are in proportion.

Sol.

Write the proportion: $7:33 :: x:66$ so $\dfrac{7}{33} = \dfrac{x}{66}$.

Cross-multiply to solve: $x = \dfrac{7 \times 66}{33}$.

Compute: $x = \dfrac{462}{33} = 14$.

Hence $x = 14$.

Unitary Method

Definition: The unitary method finds the value of one unit first, and then uses that unit value to find any required number of units. It is a two-step approach: find value of 1 unit (by dividing), then multiply to get the required number of units.

Steps

• Identify the given quantity and how many units that quantity represents.
• Divide to find the value of a single unit.
• Multiply the unit value by the required number of units.

Worked example - unitary method

Example: Sonali pays ₹9000 rent for 5 months. How much will she pay for a year (12 months)?

Sol.

Variation

Definition: Variation describes how one quantity changes when another quantity changes. Two main types are direct variation and inverse variation.

Direct variation

Definition: Two quantities vary directly when an increase in one produces a proportional increase in the other, and a decrease in one produces a proportional decrease in the other. Mathematically, $y \propto x$, or $y = kx$ for some constant $k$.

Steps to solve direct variation problems

• Find the value of one unit of the second quantity using the given information.
• Multiply that unit value by the new amount of the first quantity to find the required value.

Worked example - direct variation

Example: If 14 bags of wheat weigh 126 kg, how many bags weigh 198 kg?

Sol.

Find how many kilograms one bag weighs:
$\dfrac{126\ \mathrm{kg}}{14} = 9\ \mathrm{kg\ per\ bag}$.

To find number of bags for 198 kg, divide total kilograms by kilograms per bag:
$\dfrac{198}{9} = 22$.

Hence 22 bags weigh 198 kg.

Inverse variation

Definition: Two quantities vary inversely when as one increases, the other decreases so that their product stays constant. If $x$ varies inversely with $y$, then $x \propto \dfrac{1}{y}$ and $xy = k$ (constant).

Steps to solve inverse variation problems

• Calculate the constant product using the given values.
• Use the constant to find the unknown by dividing the constant by the new value.

Worked example - inverse variation

Example: If 5 men complete work in 8 days, how long will 1 man take to complete the same work?

Sol.

Compute the constant work measured in man-days:
$5 \times 8 = 40$ man-days.

For 1 man, days required $= \dfrac{40}{1} = 40$ days.

So 1 man will take 40 days to finish the work.

Applications and tips

• Use unit conversion first whenever units differ (e.g., kg ↔ g, hours ↔ minutes).
• Always check whether the quantities compared are of the same kind before forming a ratio or proportion.
• In sharing problems, reduce the ratio to lowest terms if it simplifies calculations.
• Use the unitary method when the given information naturally describes 'for how many units' and you need for a different number of units.
• For proportion problems, cross-multiplication is a quick and reliable method to find unknowns.

Summary

Ratio expresses how two like quantities compare; proportion states equality of two ratios; the unitary method finds the value of one unit to scale to any number of units; and variation describes direct or inverse relationships. Mastery of these ideas makes many arithmetic and algebra problems easier and helps solve practical problems in everyday life.