# Gail washed 1/6 of a stack of dishes, and Andrew washed 1/2 of the remaining dishes. 45 dishes remain. How many dishes were there before Gail and Andrew started washing them?

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by Chesnut

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by Chesnut

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Let x be the original number of dishes.

Gail washed**1/6 (x)**

Remaining dishes: 6/6 - 1/6 = 5/6

Andrew washed 1/2 of the remaining dishes:

(1/2)(5/6)(x) =**5/12 (x)**

Remaining dishes after Andrew and Gail washed some stack of dishes:

45 dishes

**Equation:**

(1/6) (x) + (5/12)(x) + 45 = x

**Multiply each term by LCD 12**:

(12)(1/6)(x) + (12)(5/12)(x) + (12)(45) = (12)(x)

2x + 5x + 540 = 12x

7x + 540 = 12x

12x - 7x = 540

5x = 540

5x/5 = 540/5

**x = 108 **

**ANSWER: There were 108 dishes before Gail and Andrew started washing them.**

Check:

**Gail** washed 1/6 of 108 = **18 dishes**

108 - 18 = 90 dishes

**Andrew **washed 1/2 of the remaining 90 dishes = **45 dishes**

Remaining dishes after Gail and Andrew washed some of the them = 45 dishes.

Add:

18 + 45 + 45 = 108

108 = 108 (true)

Gail washed

Remaining dishes: 6/6 - 1/6 = 5/6

Andrew washed 1/2 of the remaining dishes:

(1/2)(5/6)(x) =

Remaining dishes after Andrew and Gail washed some stack of dishes:

45 dishes

(1/6) (x) + (5/12)(x) + 45 = x

(12)(1/6)(x) + (12)(5/12)(x) + (12)(45) = (12)(x)

2x + 5x + 540 = 12x

7x + 540 = 12x

12x - 7x = 540

5x = 540

5x/5 = 540/5

Check:

108 - 18 = 90 dishes

Remaining dishes after Gail and Andrew washed some of the them = 45 dishes.

Add:

18 + 45 + 45 = 108

108 = 108 (true)