In math ratio and proportion we will elaborate the terms and discuss more about it in detailed explanation.
● Ratio and terms of ratio
● Properties of ratio
● Ratio in the simplest form
● Simplification of ratio
● Comparison of ratio
● Dividing the given quantity in the given ratio
● Proportion
● Continued proportion
● Examples on ratio and proportion
The ratio of two quantities 'a' and 'b' of the same kind and in the same units is a fraction \(\frac{a}{b}\) which shows that how many times one quantity is of the other and is written as a : b and is read as 'a is to b' where b ≠ 0.
In the ratio a : b, the quantities a and b are called terms of the ratio. Here, 'a' is called the first term or the antecedent and `b' is called the second term or consequent.
Example:
In the ratio 5 : 9, 5 is called the antecedent and 9 is called the consequent.
If the first term and the second term of a ratio are multiplied/divided by the same non-zero number, the ratio does not change.
● a/b = xa/xb, (x ≠ 0) So, a : b = xa : xb
● a/b = (a/x)/(b/x), (x ≠ 0) So, a : b = a/x : b/x
A ratio a : b is said to be in the simplest form if a and b have no common factor other than 1.
Example:
Express 15 : 10 in the simplest form.
Solution:
15/10
= (15 ÷ 5)/(10 ÷ 5)
= 3/2 (In this we cancelled the common factor 5)
Thus, we have expressed the ratio 15/10 in the simplest form, i.e., 3/2 and the terms 3 and 2 have common factor only 1.
Note:
● In ratio, quantities being compared must be of the same kind, otherwise the comparison becomes meaningless.
For example; comparing 20 pens and 10 apples is meaningless.
● They must be expressed in the same units.
● In a ratio, order of the terms is very important. The ratio a: b is different from b : a.
● The ratio has no units.
For example; Dozen = 12, Gross = 144, Score = 20
Decade = 10, Century = 100, Millennium = 1000
Example:
Express the following ratios in the simplest form.
(a) 64 cm to 4.8 m
(b) 36 minutes to 36 seconds
(c) 30 dozen to 2 hundred
Solution:
(a) Required ratio = 64 cm/4.8 m
= 64 cm/(4.8 × 100) cm
= 64 cm/480 m
= 64/480
= 2/15
= 2 : 15
(b) Required ratio = 36 minutes/36 seconds
= (36 × 60 seconds)/(36 seconds)
= 60/1
= 60 ∶ 1
(c) Required ratio = (30 dozen)/(2 hundred)
= (30 × 12)/(2 × 100 )
= 3/10
= 3 ∶ 10
If the terms of the ratio are expressed in fraction form; then find the Least Common Multiple of the denominators of these fractions.
Now, multiply each fraction by the L.C.M. The ratio is simplified.
Example:
Simplify the following ratios.
(a) ⁵/₂ ∶ ³/₈ ∶ ⁴/₉
(b) 2¹/₇ ∶ 3²/₅
Solution:
(a) The L.C.M. of 2, 8, 9 = 2 × 2 × 2 × 3 × 3
= 8 × 9
= 72
Now, multiplying each fraction by the L.C.M.
5/2 × 72 = 160 3/8 × 72 = 27 4/9 × 72 = 32
So, the ratio becomes 160 : 27 : 32
(b) 2¹/₇ ∶ 3²/₅
= 15/7 : 17/5 (Here, we have used (a/b)/(c/d) = \(\frac{a}{b}\) × \(\frac{d}{c}\))
= 15/7 × 5/17
= 75/119
So, the ratio becomes 75 : 119
Ratios can be compared as fractions. Convert them into equivalent ratios as we convert the given fractions into equivalent fractions and then compare.
Example:
Which ratio is greater?
2¹/₃ ∶ 3¹/₂, 2.5 : 3.5, 4/5 ∶ 3/2
Solution:
Simplifying the given 3 ratios
2¹/₃ ∶ 3¹/₂ = ⁷/₃ ∶ ⁷/₂ = ⁷/₃ ÷ ⁷/₂ = ⁷/₃ × ²/₇ = ²/₃
2.5 : 3.5 = ²⁵/₃₅ = ⁵/₇
⁴/₅ : ³/₂ = ⁴/₅ × ²/₃ = ⁸/₁₅
²/₃, ⁵/₇, ⁸/₁₅
L.C.M. of 3, 7, 15 = 105
²/₃ = (2 × 35)/(3 × 35) = ⁷<span style='font-size: 50%'>/₁₀₅,
⁵/₇ = (5 × 15)/(7 × 15) = ⁴⁵/₁₀₅,
⁸/₁₅ = (8 × 7)/(15 × 7) = ⁵⁶/₁₀₅
\(\frac{70}{105}\) > \(\frac{56}{105}\) > \(\frac{45}{105}\)
Therefore, ²/₃ > ⁸/₁₅ > ⁵/₇
Therefore, 2¹/₃ ∶ 3¹/₂ > 4/5 ∶ 3/2 > 2.5 : 3.5
If 'p’ is the given quantity to be divided in the ratio a : b, then add the terms of the a ratio, i.e., a + b, then the 1ˢᵗ part = {a/(a + b)} × p and 2ⁿᵈ part {b/(a + b)} × p
Example:
Divide $290 among A, B, C in the ratio 1¹/₂, 1¹/₄ and 7/8.
Solution:
Given ratios = ³/₂ : ⁵/₄ : 7/8.
The L.C.M. of 2, 4, 8 is 8.
So we have ³/₂ × 8 : ⁵/₄ × 8 ∶ 7/8 × 8 = 12 ∶ 10 : 7
Therefore, Share of A = \(\frac{12}{29}\) × 290 = $120
Share of B = \(\frac{10}{29}\) × 290 = $100
Share of C = \(\frac{7}{29}\) × 290 = $70
We have already learnt that statement of equality of ratios is called proportion, if four quantities a, b, c, d are in proportion, then a : b = c : d or a : b : : c : d (: : is the symbol used to denote proportion).
⇒ \(\frac{a}{b}\) = \(\frac{c}{d}\)
⇒ a × d = b × c
⇒ ad = bc
Here a, d are called the extreme terms in which a is called the first term and d is called the fourth term and b, c are called the mean terms in which b is called the second term and c is called the third term.
Thus, we say, if product of mean terms = the product of extreme terms, then the terms are said to be in proportion.
Also, if a : b :: c : d, then d is called the fourth proportional of a, b, c.
The three quantities a, b, c are said to be in continued proportion if a : b :: b : c
⇒ \(\frac{a}{b}\) = \(\frac{b}{c}\)
⇒ a × c = b²
⇒ b² = ac
⇒ b = √ac
Here, b is called the mean proportional of a and c. The square of middle term is equal to the product of 1ˢᵗ term and 3ʳᵈ term.
Also, if a : b :: b : c, then c is called the third proportional of a, b.
Example:
Determine if the following are in proportion.
(a) 6, 12, 24
(b) 1²/₃, 6¹/₄, ⁴/₉, ⁵/₃
Solution:
(a) Here, product of first term and third term = 6 × 24 = 144 and square of middle term = (12) ² = 12 × 12 = 144
(b) 1²/₃, 6¹/₄, ⁴/₉, ⁵/₃
Here, a = 1²/₃ b = 6¹/₄ c = ⁴/₉ d = ⁵/₃
a : b = 1²/₃ : 6¹/₄ c : d = ⁴/₉ : ⁵/₃
= ⁵/₃ ∶ ²⁵/₄ = (4/9)/(5/3)
= (5/3)/(25/4) = 4/9 × 3/5
= 5/3 × 4/25 = 4/3 × 1/5
= 4/15 = 4/15
Since, a : b = c : d
Therefore, 1²/₃, 6¹/₄, ⁴/₉, ⁵/₃ are in proportion.
Follow the examples on ratio and proportion then, practice the problems given in the worksheet.
● Ratio and Proportion
Worked out Problems on Ratio and Proportion
Practice Test on Ratio and Proportion
● Ratio and Proportion - Worksheets
Worksheet on Ratio and Proportion
8th Grade Math Practice
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