**Problem 1:**

The original price of a shirt was $20. It was decreased to $15 . What is the percent decrease of the price of this shirt.

**Solution to Problem 1:**

The absolute decrease is

20 - 15 = $5

The percent decrease is the absolute decrease divided by the the original price (part/whole).

percent decease = 5 / 20 = 0.25

Multiply and divide 0.25 to obtain percent.

percent decease = 0.25 = 0.25 * 100 / 100 = 25 / 100 = 25%

**Problem 2:**

Mary has a monthly salary of $1200. She spends $280 per month on food. What percent of her monthly salary does she spend on food?

**Solution to Problem 2:**

The part of her salary that is spent on food is $280 out of her monthly salary of $1200

percent = part / whole = 280 / 1200 = 0.23 (rounded to 2 decimal places)

Multiply and divide 0.23 by 100 to convert in percent

percent = 0.23 * 100 / 100 = 23 / 100 = 23%

**Problem 3:**

The price of a pair of trousers was decreased by 22% to $30. What was the original price of the trousers?

**Solution to Problem 3:**

Let x be the original price and y be the absolute decrease. If the price was decreased to $30, then

x - y = 30

y is given by

y = 22% of x = (22 / 100) * x = 0.22 x

Substitute y by 0.22 x in the equation x - y = 30 and solve for x which the original price.

x - 0.22 x = 30

0.78 x = 30

x = $38.5

Check the solution to this problem by reducing the origonal price found $38.5 by 22% and see if it gives $30.

**Problem 4:**

The price of an item changed from $120 to $100. Then later the price decreased again from $100 to $80. Which of the two decreases was larger in percentage term?

**Solution to Problem 4:**

First decrease in percent

part / whole = (120 - 100) / 120 = 0.17 = 17%

Second decrease in percent

part / whole = (100 - 80) / 100 = 0.20 = 20%

The second decrease was larger in percent term. The part were the same in both cases but the whole was smaller in the second decrease.

**Problem 5:**

The price of an item decreased by 20% to $200. Then later the price decreased again from $200 to $150. What is the percent of decrease from the original price to the final price of $150?

**Solution to Problem 5:**

We first need to find the original price x. The first decrease gives

x - 20% x = 200

0.8 x = 200

x = 200 / 0.8 = 250

The percentage decrease fro the original price 250 to 150 is given by

part / whole = (250 - 150) / 250 = 0.4 = 40%

**Problem 6:**

A number increases from 30 to 40 and then decreases from 40 to 30. Compare the percent of increase from 30 to 40 and that of the decrease from 40 to 30.

**Solution to Problem 6:**

Percent increase from 30 to 40 is given by

(40 - 30) / 30 = 10 / 30 = 0.33 = 33% (2 significant digits)

Percent decrease from 40 to 30 is given by

(40 - 30) / 40 = 0.25 = 25%

In absolute term, the percent decrease is less than the percent increase.

**Problem 7:**

A family had dinner in a restaurant and paid $30 for food. They also had to pay 9.5% sale tax and 10% for the tip. How much did they pay for the dinner?

**Solution to Problem 7:**

They paid for food, sales tax and tip, hence

total paid = $30 + 9.5% * 30 + 10% * 30 = $35.85

**Problem 8:**

A shop is offering discounts on shirts costing $20 each. If someone buys 2 shirts, he will be offered a discount of 15% on the first shirt and another 10% discount on the reduced price for the second shirt. How much would one pay for two shirts at this shop?

**Solution to Problem 8:**

The reduced price for the first shirt

20 - 15% * 20 = $17

The reduced price for the second shirt. The 10% discount will be on the already reduced price, hence the price of the second shirt is given by

17 - 10% * 17 = $15.3
The total cost for the two shirts is

17 + 15.3 = $32.3

**Problem 9:**

Smith invested $5000 for two years. For the first year, the rate of interest was 7% and the second year it was 8.5%. How much interest did he earn at the end of the two year period?

**Solution to Problem 9:**

Interest at the end of the first year

7% * 5000 = $350

Interest at the end of the second year

8.5% * (5000 + 350) = $454.75

Total interest at the end of the two year period is

$350 + $454.75 = $804.75

**Problem 10:**

Janette invested $2000 at 5% compounded annually for 5 years. How much interest did she earn at the end of the 5 year period?

**Solution to Problem 10:**

At the of the first year, she has the principal plus the interest on the principal

P1 = 2000 + 5% * 2000 = 2000(1 + 5%)

At the of the second year, she has the principal P1 plus the interest on P1

P2 = P1 + 5% * P1 = P1(1 + 5%)

Substitute P1 by 2000(1 + 5%) found above to find

P2 = 2000 * (1 + 5%)^{ 2}

Continuing with this process, it can easily be shown that a the end of the 5th year, the principal is given by

P5 = 2000 * (1 + 5%)^{ 5}

= 2000 * (1 + 0.05) = $2552.56

The interest earned at the end of 5 years is

$2552.56 - $2000 = $552.56

**Problem 11:**

Tom borrowed $600 at 10% per year, simple interest, for 3 years.
How much did he have to repay (principal + interest) at the
end of the 3 year period?

**Solution to Problem 11:**

The interest to pay is given by

Interest = 600 * 10% * 3 = $180

Total to repay

600 + 180 = $780

**Problem 12:**

Out of a world population of approximately 6.6 billion, 1.2 billion people live in the richer countries of Europe, North America, Japan and Oceania and is growing at the rate of 0.25% per year, while the other 5.4 billion people live in the lees developed countries and is growing at the rate of 1.5%. What will be the world population in 5 years if we assume that these rates of increase will stay constant for the next 5 years. (round answer to 3 significant digits)

**Solution to Problem 12:**

Let us first calculate the population PR in 5 years in the richer countries

PR = (1.2 + 0.25% * 1.2) = 1.2(1 + 0.25%) after one year

PR = 1.2(1 + 0.25%) + 0.25% * 1.2(1 + 0.25%)

= 1.2(1 + 0.25%)^{ 2}after two years

Continue with the above and after 5 years, PR will be

PR = 1.2(1 + 0.25%)^{ 5} after 5 years

Similar calculations can be used to find the population PL in less developed countries after 5 years.

PL = 5.4(1 + 1.5%)^{ 5} after 5 years

The world population P after 5 years will be

P = PR + PL = 1.2(1 + 0.25%)^{ 5} + 5.4(1 + 1.5%)^{ 5} = 7.03 billion.

**Problem 13:**

Cassandra invested one part of her $10,000 at 7.5% per year and the other part at 8.5% per year. Her income from the two investment was $820. How much did she invest at each rate?

**Solution to Problem 13:**

Let x and y be the amount invested at 7.5% and 8.5% respectively

Income = $820 = 7.5% * x + 8.5% * y

The total amount invested is also known

10,000 = x + y

Solve the system of the equations to find x and y.

x = $3000 and y = $7000

As a practice check that 7.5% of $3000 and 8.5% of $7000 gives $820.

**Problem 14:**

The monthly salary S of a shop assistant is the sum of a fixed salary of $500 plus 5% of all monthly sales. What should the monthly sales be so that her monthly salary reaches $1500?

**Solution to Problem 14:**

Let S be the total monthly salary and x be the monthly sales, hence

S = 500 + 5% * x

Find sales x so that S = 1500, hence

1500 = 500 + 5% * x = 500 + 0.05 x

Solve for x

x = (1500 - 500) / 0.05 = $20000

**Problem 15:**

A chemist has a 20% and a 40% acid solutions. What amount of each solution should be used in order to make 300 ml of a 28% acid solution?

**Solution to Problem 14:**

Let x be the solution at 20% and y be the solution at 40%, hence

x + y = 300 ml

We now write an equation that expresses that the total acid in the final 300 ml is equal to the sum of the amounts of acid in x and y

28% * 300 = 20% * x + 40% * y

Solve the above system of equations to find

x = 180 and y = 120

**Problem 16:**

What percent of the total area of the circular disk is colored red?

**Solution to Problem 16:**

Total area of disk

Ad = pi * r^{ 2}

Angle t in radians of central angle of red sector

t = (360-120)* pi / 180 = (4/3) pi

Area of red sector

As = (1/2) t * r^{ 2}

Percentage of total area in red

P = [ (1/2) t * r^{ 2} ] / [ pi * r^{ 2} ]

= 4 / 6 = 66.7% (3 significant digits)

THINK: compare 66.7% to 240 / 360, why are they equal?

**Problem 17:**

What percent of the total area of the rectangle is colored red?

**Solution to Problem 17:**

Total area of rectangle

Ar = L * W

Area of triangle

At = (1/2) base * height = (1/2) [ L * (1/2) W ]

Percentage of area in red

P = (1/2) [ L * (1/2) W ] / [L*W] = 1/4

= 25%

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