Ratio and Proportion

This guide breaks down the Ratio and Proportion PDF into simple explanations, key formulas, solved examples, a cheatsheet, practice questions (with answers), and visual aids for better understanding.


1. Ratio: Definition and Basics

  • Ratio compares two quantities by division.
  • Written as a:b a : b , which means ab \frac{a}{b} .
  • Antecedent: First term (a), Consequent: Second term (b).

Example: Ratio 5:9 means antecedent = 5, consequent = 9.

Visual: Ratio Bar

|-----5-----|-----9-----|
   a (5)       b (9)

2. Proportion: Definition

  • Proportion is the equality of two ratios.
  • If a:b=c:d a : b = c : d , written as a:b::c:d a : b :: c : d .
  • Extremes: a and d; Means: b and c.
  • Rule: Product of means = Product of extremes bร—c=aร—d b \times c = a \times d

3. Types of Proportion

  • Third Proportion: If a:b=b:c a : b = b : c , then c is the third proportion to a and b.
  • Fourth Proportion: If a:b=c:d a : b = c : d , then d is the fourth proportion to a, b, c.
  • Mean Proportion: Mean proportional between a and b is ab \sqrt{ab} .

4. Compounded Ratio

  • Multiply antecedents and consequents of two or more ratios.
  • For (a:b),(c:d),(e:f) (a:b), (c:d), (e:f) , compounded ratio is ace:bdf ace : bdf .

5. Key Formulas Cheatsheet

ConceptFormula/Rule
Ratio a:b a:b ab \frac{a}{b}
Proportion a๐Ÿ…ฑยฎ:c:d a๐Ÿ…ฑ๏ธ:c:d bร—c=aร—d b \times c = a \times d
Third Proportiona:b=b:cโ€…โ€ŠโŸนโ€…โ€Šc=b2a a : b = b : c \implies c = \frac{b^2}{a}
Fourth Proportiona:b=c:dโ€…โ€ŠโŸนโ€…โ€Šd=bร—ca a : b = c : d \implies d = \frac{b \times c}{a}
Mean Proportionab \sqrt{ab}
Compounded Ratioace:bdf ace : bdf

6. Solved Examples

Q1. Combine Two Ratios

If A:B=2:3 A:B = 2:3 and B:C=5:7 B:C = 5:7 , find A:B:C A:B:C .

Solution:

  • Make B common: A:B=2:3 A:B = 2:3 , B:C=5:7 B:C = 5:7 Multiply A:B A:B by 5, B:C B:C by 3: A:B=10:15 A:B = 10:15 , B:C=15:21 B:C = 15:21 So, A:B:C=10:15:21 A:B:C = 10:15:21

Q2. Compounded Ratio

Find the compound ratio of 17:23, 115:153, 18:25.

Solution:

  • Multiply antecedents: 17ร—115ร—18 17 \times 115 \times 18
  • Multiply consequents: 23ร—153ร—25 23 \times 153 \times 25
  • =(17ร—115ร—18):(23ร—153ร—25)=2:5 = (17 \times 115 \times 18) : (23 \times 153 \times 25) = 2:5

Q3. Fourth Proportion

If 3:27::5:? 3:27 :: 5:?

Solution:

  • 3/27=5/xโ€…โ€ŠโŸนโ€…โ€Šx=(5ร—27)/3=45 3/27 = 5/x \implies x = (5 \times 27)/3 = 45

Q4. Third Proportion

What is the third proportion to 17.9 and 16.8?

Solution:

  • 17.9:x=16.8:xโ€…โ€ŠโŸนโ€…โ€Šx=16.8217.9=15.76 17.9:x = 16.8: x \implies x = \frac{16.8^2}{17.9} = 15.76

Q5. Mean Proportion

Find the mean proportional between 14 and 15.

Solution:

  • 14ร—15=210โ‰ˆ14.5 \sqrt{14 \times 15} = \sqrt{210} \approx 14.5

Q6. Mixed Proportion Problem

Mean proportional of 4 and 36 is a a ; third proportional of 18 and a a is b b . Find the fourth proportional of b,12,14 b, 12, 14 .

Solution:

  • Mean proportional of 4 and 36: a=4ร—36=12 a = \sqrt{4 \times 36} = 12
  • Third proportional of 18 and 12: b=12218=8 b = \frac{12^2}{18} = 8
  • Fourth proportional of 8, 12, 14: 8/12=14/xโ€…โ€ŠโŸนโ€…โ€Šx=21 8/12 = 14/x \implies x = 21

7. Practice Questions with Answers

Q7. Coin Ratio Problem

A bag has coins of Rs. 1, 50 paise, and 25 paise in the ratio 5:9:4. If the total number of coins is 72, what is the worth of the bag?

Solution:

  • Rs. 1 coins: 5/18ร—72=20 5/18 \times 72 = 20
  • 50 paise coins: 9/18ร—72=36 9/18 \times 72 = 36
  • 25 paise coins: 4/18ร—72=16 4/18 \times 72 = 16
  • Total value: (20ร—1)+(36ร—0.5)+(16ร—0.25)=20+18+4=Rs.42 (20 \times 1) + (36 \times 0.5) + (16 \times 0.25) = 20 + 18 + 4 = Rs. 42

Q8. Variable Ratio Problem

If 18:13.5:16:x 18:13.5:16:x and (x+y):y:18:10 (x+y):y:18:10 , what is the value of y y ?

Solution:

  • x=(16ร—13.5)/18=12 x = (16 \times 13.5)/18 = 12
  • (12+y):y=9:5โ€…โ€ŠโŸนโ€…โ€Š5(12+y)=9yโ€…โ€ŠโŸนโ€…โ€Š4y=60โ€…โ€ŠโŸนโ€…โ€Šy=15 (12+y):y = 9:5 \implies 5(12+y) = 9y \implies 4y = 60 \implies y = 15

Q9. Share Division

Mr. Raj divides Rs. 1573 so that 4 times the 1st share, 3 times the 2nd, and 2 times the 3rd are equal. Find the value of the 2nd share.

Solution:

  • Let shares be A,B,C A, B, C such that 4A=3B=2C 4A = 3B = 2C
  • A:B:C=1/4:1/3:1/2=3:4:6 A:B:C = 1/4:1/3:1/2 = 3:4:6
  • Total shares = 13 parts; 2nd share = (4/13)ร—1573=Rs.484 (4/13) \times 1573 = Rs. 484

8. Visuals & Graphics

A. Ratio Pie Chart Example

If A:B:C = 2:3:5

[ 20% ] [ 30% ] [ 50% ]
   A        B       C

B. Proportion Bar

|---a---|---b---| = |---c---|---d---|

If a:b=c:d a:b = c:d , then the lengths are proportional.


9. Quick Reference Table

ConceptFormula/Rule
Ratioa:b=ab a:b = \frac{a}{b}
Proportiona๐Ÿ…ฑยฎ:c:dโ€…โ€ŠโŸนโ€…โ€Šad=bc a๐Ÿ…ฑ๏ธ:c:d \implies ad = bc
Mean Proportionab \sqrt{ab}
Third Proportionc=b2a c = \frac{b^2}{a}
Fourth Proportiond=bร—ca d = \frac{b \times c}{a}
Compounded Ratioace:bdf ace : bdf

10. Tips & Tricks

  • To combine ratios, make the common term equal.
  • For mean proportion, always use the square root.
  • For compounded ratio, multiply all antecedents and all consequents.
  • For share division, convert conditions into ratios.

With these explanations, formulas, solved examples, cheatsheet, practice questions, and visuals, youโ€™re ready to master Ratio and Proportion for any exam!

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