SOLVING A REAL WORLD PROBLEM USING THE DISTRIBUTIVE PROPERTY

Problem 1 :

A family had their bill at a restaurant reduced by $7.50 because of a special discount. They left a tip of $8.90, which was 20% of the reduced amount. How much was their bill before the discount ?

Solution : 

Step 1 :

Let 'x' be the bill amount before the discount. 

Then, amount to be paid after the discount $7.50 is 

=  x - 7.5

So, the reduced amount is (x - 7.5).

Step 2 : 

20% of the reduced amount is the tip $8.90.

So, we have 

0.2(x - 7.5)  =  8.9

0.2x - 1.5  =  8.9

Step 3 : 

Use inverse operations to solve the equation.

Add 1.5 to both sides.

0.2x  =  10.4

Since we have one digit after the decimal on both sides, multiply both sides by 10 to avoid decimal.

10(0.2x)  =  10(10.4)

2x  =  104

Divide both sides by 2. 

2x / 2  =  104 / 2

x  =  52

So, the family’s bill before the discount was $52.00.

Justify and Evaluate :

$52.00 - $7.50  =  $44.50

0.2($44.50)  =  $8.90. 

This is the amount given by the family as a tip. So, the answer is reasonable. 

Problem 2 :

The members of a club spend 8% of their budget on entertainment. Their total budget for this year is $2,000 more than last year, and this year they plan to spend $3,840 on entertainment. What was their total budget last year ?

Solution : 

Step 1 :

Let 'x' be the total budget last year. 

Then, total budget for this year is

=  x + 2000

Step 2 : 

This year, they plan to spend $3,840 on entertainment. 

Usually, they spend 8% of their budget on entertainment. 

So, we have

8% of the total budget for this year  =  3,840

0.08(x + 2000)  =  3840

0.08x + 160  =  3840

Step 3 : 

Use inverse operations to solve the equation.

Subtract 160 from each side. 

0.08x  =  3680

Since we have two digits after the decimal in 0.08x, multiply both sides by 100 to avoid decimal.

100(0.08x)  =  100(3680)

8x  =  368000

Divide each side by 8. 

8x / 8  =  368000 / 8

x  =  46000

So, the total budget for the last year was $46,000. 

Justify and Evaluate :

$46,000 + $2,000  =  $48,000

0.08($48,000)  =  $3,840 

This is the amount planned on entertainment for this year. So, the answer is reasonable. 

Problem 3 :

Solve 

y(y - 12)  + y(y + 2) + 25 = 2y(y + 5) - 15

Solution :

y² - 12y + y² + 2y + 25 = 2y² + 10y - 15

2y² - 10y + 25 = 2y² + 10y - 15

-10y - 10y = -15 - 25

-20y = -40

y = 40/20

y = 2

Problem 4 :

Marta has $6000 to invest. She put x dollars of this money into a savings account that earns 3% per year., with the rest, she buys a certificate of deposit that earns 6% per year. 

i) Write an equation for the total amount of money T Marta will have after one year. 

ii)  Suppose at the end of one year, Marta has a total of $6315. How much money did Marta invest in each account ?

Solution :

Total amount Marta has = $6000

Money earns by Marta in her savings = 3% of x

Rest of the money = 6000 - x

Amount earns by deposit = 6% of (6000 - x)

i) Amount earned after one year

= 3% of x + 6% of (6000 - x)

ii) Total earning after a year = 6315

3% of x + 6% of (6000 - x) = 6315 - 6000

0.03x + 0.06(6000 - x) = 315

0.03x + 360 - 0.06x = 315

-0.03x = 315 - 360

-0.03x = -45

x = 45/0.03

x = 1500

So, amount she is investing is $1500.

Problem 5 :

Mr Smith's American History class will take taxis from their hotel in Washington D.C to the Lincoln Memorial. The fare is $2.75 for the first mile and $1.25 for each additional mile. If the distance is m miles and t taxis are needed, write the expression for the cost to transport the group.

Solution :

Distance covered = x miles

Cost spent for the first mile = $2.75

For each additional miles, the cost spent = $1.25

= 2.75 + 1.25(m - 1)

= 2.75 + 1.25m - 1.25

= 1.5 + 1.25m

Problem 6 :

Laura is making baskets of apples and oranges for homeless shelters. She want to place a total of 10 pieces of fruit in each basket. Apples cost 25 cent each and oranges cost 20 cents each.

a) If a represents the number of apples Laura uses, write a polynomial model in simplest form for the total amount of money T Laura will spend on the  fruit of each basket.

b)  If Laura uses 4 apples in each basket, find the total cost of fruit.

Solution :

Total number of pieces of fruit = 10

Cost of each apple = $0.25

Cost of each orange = $0.20

Let x be the number of apples in the basket, then number of oranges will be 10 - x

i)  T = 0.25x + (10 - x)0.20

ii) When x = 4

T = 0.25x + (10 - x)0.20

T = 0.25x + 2 - 0.20x

= 0.05x + 2

Applying 4,

= 0.05(4) + 2

= 0.2 + 2

= 2.2

Then total cost of fruits is $2.2

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