Proportion, Merging Ratios & Continuous Ratios

What is Proportion?

When two ratios are equal to each other, they are in proportion.

a/b = c/d → written as a : b :: c : d

The symbol :: is the sign of proportion.

Finding the missing term

Cross-multiply: a × d = b × c

Example 1: 16 : 5 :: 48 : x

• 16/5 = 48/x → 16x = 48 × 5 = 240 → x = 15

Example 2: 17 : P :: 68 : 32

• 17/P = 68/32 = 4/P → wait: 68/32 simplified.
• 17 × 32 = 68 × P → P = (17 × 32)/68 = 8

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Merging Two Ratios into a Continuous Ratio

Given A:B and B:C separately, find A:B:C.

Method: Make the common term (B) equal in both ratios.

Example: A:B = 7:4, B:C = 5:6, C:D = 9:2. Find A:D.

	A	B	C	D
A:B	7	4
B:C		5	6
C:D			9	2

Step 1: Equalize B → A:B = 7×5 : 4×5 = 35 : 20, B:C = 5×4 : 6×4 = 20 : 24

Step 2: Now equalize C → Currently A:B:C = 35:20:24. C:D = 9:2.

• Multiply A:B:C by 9: 315 : 180 : 216
• Multiply C:D by 24: 216 : 48

Result: A : B : C : D = 315 : 180 : 216 : 48

Simplify: divide by 3 → 105 : 60 : 72 : 16... let's verify: 315/3=105, 180/3=60, 216/3=72, 48/3=16. GCD check: 105:60:72:16 → not further simplifiable.

So A : D = 105 : 16.

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Merging Ratios Using Percentage

Example: A is 45.45% less than B. C is 13.33% more than A. Find B:C and how much % C is less than B.

Step 1: A:B from "A is 45.45% less than B"

• 45.45% = 5/11 → A = B − 5B/11 = 6B/11
• A : B = 6 : 11

Step 2: A:C from "C is 13.33% more than A"

• 13.33% = 2/15 → C = A + 2A/15 = 17A/15
• A : C = 15 : 17

Step 3: Merge by equalizing A:

• A:B = 6:11 → × 5 → 30:55
• A:C = 15:17 → × 2 → 30:34

A : B : C = 30 : 55 : 34

B : C = 55 : 34. Difference = 21. C is less than B by: 21/55 × 100 = 38.18%

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Finding Ratios from Equations

Type 1: ax = by = cz → find x : y : z

Method: Set the common value = LCM of a, b, c. Then solve.

Example: 9x = 6y = 4z and z = 72. Find x.

Method 1 (Direct): 9x = 4z = 4(72) = 288 → x = 288/9 = 32

Method 2 (LCM): Set 9x = 6y = 4z = LCM(9,6,4) = 36

• x = 36/9 = 4, y = 36/6 = 6, z = 36/4 = 9
• x : y : z = 4 : 6 : 9

Example: 5a = 4b = 10c and a + b + c = 220. Find b.

• Set 5a = 4b = 10c → use LCM(5,4,10) = 20
• a = 20/5 = 4, b = 20/4 = 5, c = 20/10 = 2
• a : b : c = 4 : 5 : 2
• Total = 11x = 220 → x = 20
• b = 5(20) = 100

Example: (3/8)P = (4/9)Q = (5/6)R and P + Q + R = 367. Find R.

Method 1: Set each = 60 (LCM of 8,9,6... actually let's pick a convenient number)

• (3/8)P = 60 → P = 160
• (4/9)Q = 60 → Q = 135
• (5/6)R = 60 → R = 72

P : Q : R = 160 : 135 : 72

Total = 367 parts. Check: 160 + 135 + 72 = 367. So multiplier = 1. R = 72

Method 2 (Cross-product):

• P : Q : R = (4/9 × 5/6) : (3/8 × 5/6) : (3/8 × 4/9)
• = 20/54 : 15/48 : 12/72
• Simplify to get same ratio: 160 : 135 : 72

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Ratio from (a−b) and (a+b)

Example: (a−b) : (a+b) = 3 : 8 and a + b = 176. Find a.

• (a−b)/(a+b) = 3/8
• Cross multiply: 8(a−b) = 3(a+b)
• 8a − 8b = 3a + 3b → 5a = 11b → a/b = 11/5
• a = 11x, b = 5x → a + b = 16x = 176 → x = 11
• a = 11(11) = 121

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Practice Problems

#	Problem	Answer
1	16:5 :: 48:x. Find x	15
2	A:B = 7:4, B:C = 5:6, C:D = 9:2. Find A:D	105:16
3	9x = 6y = 4z, z = 72. Find x	32
4	5a = 4b = 10c, a+b+c = 220. Find b	100
5	(3/8)P = (4/9)Q = (5/6)R, P+Q+R = 367. Find R	72
6	(a−b):(a+b) = 3:8, a+b = 176. Find a	121