Exercise 68 Page 100

Practice makes perfect

The problem says that Tim can paint a house in 6 days while Tara does it in 3. Then it asks us to find what fraction of the house each would paint in 1 day. Notice that 1 day is one sixth of the time Tim needs.

Tim: 1day/6days = 1/6 Therefore, he would paint 16 of the house in one day. Now let's find the fraction of the house Tara would paint in one day. Tara:1day/3days = 1/3 We can see she would paint 13 of the house in one day.

Now, we want to find what fraction of the house each person would paint in d days. Since Tim paints 16 of the house per day, in d days he would have painted d times that quantity.

1/6 * d = d/6 Tim paints d6 of the house in d days. Following the same reasoning for Tara, she would paint d3 of the house.

Now we are asked what fraction of the house they would paint together in 1 day and in d days. For this we can add the fractions of the house they would paint individually, as each of them will be painting at the same pace they would be painting alone. Let's find this for 1 day first.

1/6+1/3Now let's add these fractions. Recall that in order to add fractions together they must have the same denominator.

1/6 + 1/3

ExpandFrac

a/b=a * 2/b * 2

1/6 + 1* 2/3 * 2

Multiply

1/6 + 2/6

AddFrac

Add fractions

3/6

ReduceFrac

a/b=.a /3./.b /3.

3÷ 3/6÷ 3

Div

Divide by 3

1/2

Hence, if painting together they will paint 12 of the house in 1 day. Therefore, if each day painting together they paint 12 of the house, in d days they will paint d times that fraction. 1/2 * d = d/2 Thus, painting together they paint d2 of the house in d days.

Finally, we need to write an equation that we can solve to find the days they would need to paint the whole house working together. Since d2 is the part of the house they paint together in d days, they will finish painting one house when that expression is equal to 1.

d/2 = 1 We can solve this equation for d using inverse operations and following the Properties of Equality.

d/2= 1

MultEqn

LHS * 2=RHS* 2

d/2 * 2= 1 * 2

FracMultDenomToNumber

a/2* 2 = a

d = 1* 2

Multiply

d = 2

Hence, they would need 2 days to paint the whole house if working together.