SOLUTION: WORD PROBLEM: PLS. HELP ME SOLVE THIS PROBLEM. THANK YOU. PROBLEM 1. When A and B both work, they can paint a certain house in 8 days. Also, they could paint this house if A wo

Question 350802: WORD PROBLEM:
PLS. HELP ME SOLVE THIS PROBLEM. THANK YOU.
PROBLEM 1. When A and B both work, they can paint a certain house in 8 days. Also, they could paint this house if A worked 12 days and B worked 6 days. How long would it take to paint the house alone?
Problem 2: Workmen A and B complete a certain job if they work together in 6 days, or if A works for 12 days and B works for 3 days. How long would it take each man, alone, to complete the job?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
Problem 1:

A and B both paint the house in 8 days.
A works 12 days and B works 6 days and the house is painted in 18 days.

Let x = rate that A paints the house.
Let y = rate that B paints the house.
Let 1 = Number of Units (a painted house)

Rate * Time = Units

Working together, they can paint the house in 8 days.

Formula is:

(x + y) * 8 = 1

Working separately, they can complete the house in 18 days, with x rated person working 12 days and y rated person working 6 days.

Formula is:

(x * 12) + (y * 6) = 1

You have 2 equations that need to be solved simultaneously.

They are:

(x + y) * 8 = 1

(x * 12) + (y * 6) = 1

These equations can be re-written as:

8*x + 8*y = 1
12*x + 6*y = 1

Multiply first equation by 6 to get:

48*x + 48*y = 6

Multiply second equation by 8 to get:

96*X + 48*Y = 8

Two equations are now:

48*x + 48*y = 6
96*x + 48*y = 8

Subtract first equation from second equation to get:

48*x = 2

Divide both sides of this equation by 48 to get:

x = 2/48 = 1/24

Substitute for x in the original first equation and solve for y.

Original first equation is:

8*x + 8*y = 1

Substitute 1/24 for x to get:

8*(1/24) + 8*y = 1

Simplify to get:

1/3 + 8*y = 1

Subtract 1/3 from both sides of the equation to get:

8*y = 2/3

Divide both sides of the equation by 8 to get:

y = 2/24 = 1/12

Rate of workman A is 1/24 of the house in a day.
Rate of workman B is 1/12 of the house in a day.

Working together they take 8 days.

8 * 1/24 + 8 * 1/12 = 8/24 + 8/12 = 8/24 + 16/24 = 24/24 = 1.

First equation checks out.

Working alone, workman A takes 12 days and workman B takes 6 days.

12 * (1/24) + 6 * (1/12) = 12/24 + 6/12 = 12/24 + 12/24 = 24/24 = 1.

Second equation checks out.

Workman A would take 24 days to paint the house alone.

Workman B would take 12 days to paint the house alone.

Problem 2:

Workmen A and B complete a certain job if they work together in 6 days, or if A works for 12 days and B works for 3 days. How long would it take each man to complete the job.

Rate * Time = Units

Number of Units = 1 (the job).

Workman A rate = x
Workman B rate = y

(x + y) * 6 = 1 (first equation with workmen working together).

x*12 + y*3 = 1 (workman A works 12 days and workman B works 3 days).

Solve equations simultaneously to get the answer.

Equations can be re-written as:

6*x + 6*y = 1
12*x + 3*y = 1

Multiply second equation by 2 to get:

6*x + 6*y = 1
24*x + 6*y = 2

Subtract equation 1 from equation 2 to get:

18*x = 1

Divide both sides of the equation by 18 to get:

x = 1/18

Substitute in first equation to get:

6*(1/18) + 6*y = 1

Simplify to get:

6/18 + 6*y = 1

Simplify further to get:

1/3 + 6*y = 1

Subtract 1/3 from both sides of the equation to geyt:

6*y = 2/3

Divide both sides of the equation by 6 to get:

y = 2/18

Simplify further to get:

y = 1/9

You have rates for workman A and B as follows:

Workman A rate is 1/18 of the job in a day.

Workman B rate is 1/9 of the job in a day.

Working together, the original first equation becomes:

6 * (1/18 + 1/9) = 6 * 3/18 = 18/18 = 1.

Rates check out ok in first equation.

Working separately, the original second equation becomes:

12 * (1/18) + 3 * (1/9) = 12/18 + 3/9 = 12/18 + 6/18 = 18/18 = 1.

Rates check out ok in second equation.

Working separately:

Workman A will take 18 days to complete the job.
Workman B will take 9 days to complete the job.

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