Solve multi-step proportions

Learn how to solve multi-step proportions easily with a couple of good examples.

Quick review:

Suppose you have the following proportion:

Numerator #1 / Denominator #1 = Numerator #2 / Denominator #2 

After doing cross-multiplication, it is equivalent to 

Numerator #1 × Denominator #2 = Numerator #2 × Denominator #1

Example #1

Solve the proportion (x - 5) / 4 = (x + 3) / 6

(x - 5) / 4 = (x + 3) / 6

The proportion is equivalent to 

(x - 5) × 6 = (x + 3) × 4

(x - 5) × 6 = (x + 3) × 4

Use the distributive property

6x - 30 = 4x + 12

Subtract 4x from each side of the equation

6x - 4x - 30 = 4x - 4x + 12

2x - 30 = 0 + 12

2x - 30 = 12

Add 30 to each side of the equation

2x - 30 + 30 = 12 + 30

2x + 0 = 42

2x = 42

Divide each side by 2

2x/2 = 42/2

x = 21

Example #2

Solve the proportion 3 / (x - 4) = -5 / (4x + 1)

3 / (x - 4) = -5 / (4x + 1)

The proportion is equivalent to 

3 × (4x + 1) = -5 × (x - 4)

3 × (4x + 1) = -5 × (x - 4)

Use the distributive property

12x + 3 = -5x + 20

Add 5x to each side of the equation

12x + 5x + 3 = -5x + 5x + 20

17x + 3 = 0 + 20

17x + 3 = 20

Subtract 3 from each side of the equation

17x + 3 - 3 = 20 - 3

17x + 0 = 17

17x = 17

Divide each side by 17

17x/17 = 17/17

x = 1

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