Business Mathematics: Ratio, Proportion, and Percent - Problems Of Increase And Decrease

If a quantity increases or decreases, the amount of change is often represented as a percentage of the original quantity. Thus, if an inventory of 100 radios increases by 25%, the new quantity is 125.

Example
Don’s appliance store ordinarily stocks 250 assorted appliances. On June 1, Don noticed that his inventory increased by 12%. How many appliances did Don have in stock on June 1?
Since 12% of 250 = 0.12 × 250 = 30 appliances, Don had 280 (250 + 30) appliances in stock.

Example
A well-known department store sells a certain brand of perfume for $90 an ounce. During a 2-week promotion they reduced the price by 15%. What was the sale price of the perfume?
Since 15% of 90 = 0.15 × $90 = 13.50, the sale price was 90 – 13.50 = $76.50 per ounce.

Example
The price of a new automobile, including an % sales tax, is $16,531.06. What is the price of the car before tax?
Let x be the price of the car before sales tax is added. The tax itself is 0.085x (x). Therefore

Divide both sides of the equation by 1.085 to get x = $15,236.

Solved Problems

2.29   How much is 27% more than $500?
Solution
The increase is 27% of 500 = 0.27 × 500 = $135. The new value is 500 + 135 = $635.

2.30   If a price of $80 is increased by 140%, how much will the new price be?
Solution
The price increase = 140% of 80 = 1.40 × 80 = $112. The new price is $80 + $112 = $192.

2.31   What number is 30% less than 210?
Solution
30% of 210 = 0.30 × 210 = 63. The new number is 63 less than 210 = 210 – 63 = 147.

2.32 If the number 100 is increased by 10% and then that number is decreased by 10%, do we get the original 100 back?
Solution
No! 10% of 100 = 0.10 × 100 = 10. Adding this to 100, we get 110. Now take 10% of 110. 0.10 × 110 = 11. Subtracting this from 110 gives 99.

2.33   How much will a $650 stereo system cost if its price is increased by 15%?
Solution
15% of $650 = 0.15 × 650 = $97.50. The new price is $650 + $97.50 = $747.50.

2.34   A $750 fur jacket goes on sale at a discount of 20%. What is the sale price of the jacket?
Solution
20% of $750 = 0.20 × 750 = $150. The sale price is $750 – $150 = $600.

2.35   A gallon of paint now costs $11.80, but there will be a 15% increase in price tomorrow. How much will the actual price increase be? What will the new price of a gallon of paint be?
Solution
15% of 11.80 = 0.15 × 11.80 = 1.77. The increase is $1.77. The new price will be $11.80+ $1.77 = $13.57.

2.36   A diamond necklace costs $10,272, including a 7% sales tax. How much was the necklace before tax?
Solution
Let x = price of the necklace before tax. The tax is 7% × x.

The price before tax is $9,600.

2.37   The price of a tennis racket is $126 after a 10% reduction is taken. How much was the racket before the reduction was taken?
Solution
Let x = the price of the racket before the reduction.

The racket costs $140 before the reduction is taken.
Supplementary Problems

2.38 Find the ratio of 3 weeks to 6 days.

2.39 Find the ratio of 12 yd to 9 ft.

2.40 Distribute $18,000 in the ratio 5:7.

2.41 A stock dividend is distributed among shareholders in the ratio of shares held. If Tom, Robert, and Sally have 500, 350, and 200 shares, respectively, and their combined dividend is $5,250, how much of the dividend does each receive?

2.42 $500,000 is to be allocated by a small computer company for equipment, advertising, and salaries in the ratio 3:2:5. How much money is allocated for each?

2.43 A family budget allocates money for food, housing, and entertainment in the ratio 4:9:1. If the total amount of money allocated for these expenses is $22,750, how much is to be spent on each item?

2.44 Three physicians invested $150,000, $200,000, and $250,000, respectively, in a medical practice. Their profit for the year was $1,200,000. If they divide the profits in the same ratio as their investments, how much does each receive?

2.45 Determine the value of x if x : 27 = 4:3.

2.46 Determine the value of y if 5:y = 20:3.

2.47 How many pounds of hard candies should be mixed with 30 lb of soft candies if their weight ratio is to be 3:2?

2.48 If a car can travel 150 mi on 8 gal of gas, how many gallons will be needed for a 675 mi trip?

2.49 The property tax in a small town is $17 per $100 of assessed valuation. How much is a house assessed for if its property tax is $2,975?

2.50 A philanthropist gives $140,000 to the heart foundation and the cancer society in the ratio 4:3. How much money does each charity receive?

2.51 The contestants in a boxing-match will receive prizes in the ratio 7:4; the winner gets the larger amount. If the total amount to be given out is $550,000 how much will each boxer receive?

2.52 A cookie recipe uses 2 teaspoons of butter for every 50 cookies made. How many teaspoons of butter will be needed if Lindsay wishes to make 225 cookies for a cookie sale?

2.53 The ratio of red balloons to blue balloons in a balloon display is 5:3. How many balloons of each color are present in a display consisting of 360 balloons?

2.54 y is directly proportional to x. y = 30 when x= 5.
  (a)   Find the constant of proportionality.
  (b)   Determine the value of y when x = 15.

2.55 y is inversely proportional to x. y = 30 when x = 5.
  (a)   Find the constant of proportionality.
  (b)   Determine the value of y when x = 15.

2.56 The amount of commission Bruce Sawyer, a salesman, receives is directly proportional to his sales. If Bruce receives $3,200 when he sells $50,000 worth of merchandise, how much will he receive if his sales for the month are $70,000?

2.57 The amount of money Joan earns is directly proportional to the number of hours she works. One week she worked 35 hours and earned $245. How much will she earn if she works 50 hours the following week?

2.58 It takes 20 workers a total of 30 days to pave a highway. How many days would it take 15 workers to do the same job?

2.59 Express as a fraction and simplify: (a) 1.25%, (b) 12.5%, (c) 125%.

2.60 Express as a percentage: (a) 9/50, (b) 11/20, (c) 8/25.

2.61 Express as a decimal: (a) 59%, (b) 5.9%, (c) 0.59%.

2.62 Compute (a) 18% of 90, (b) 30% of 600.

2.63 (a) 30% of what number is 18? (b) 45% of what number is 135?

2.64 How much tax will be collected on the purchase of a $450 TV set if the sales tax rate is 7%?

2.65 How much tax will be collected on a new car costing $25,100 if the sales tax rate is 8%?

2.66 How much tax is charged on a $2,700 item if the tax rate is 5% on the first $2,000 and 7% on the amount over $2,000?

2.67 How much tax is charged on a $37,000 automobile if the basic tax rate is %on the first $30,000 and 11% on the excess?

2.68 Suppose that the income tax rate is 20% on the first $30,000 of income, 15% on all income above $30,000 but not exceeding $50,000, and 10% on income above $50,000. How much tax will Bob have to pay if his income was $72,300?

2.69 There is a 7% sales tax on all purchases made in Eastport. If $17.50 tax was charged, how much did the item originally cost?

2.70 If the total price of a cordless telephone, including % sales tax, is $103.20
  (a)   What was the pretax price of the phone?'
  (b)   How much tax was charged on the phone?

2.71 To rent a car for one day with unlimited mileage, you must pay a $2.00 highway tax and a 12% tax on the rental. If it costs $52.40 to rent a car, what was the base price of the rental?

2.72 If there is an % sales tax on new automobiles and a new Cadillac costs a total of $34,640 including tax, what was the pretax price of the automobile?

2.73 What is the tax rate if the tax on a $750 computer is $52.50?

2.74 To buy a $3,000 fur coat, a customer has to pay $3,255 including taxes. What is the tax rate in the town where the coat was purchased?

2.75 What number is 30% less than 90?

2.76 The retail price of a dress is 115 % of wholesale. What is the retail price of a dress which wholesales for $140?

2.77 If the number 1000 is increased by 20% and then that number is decreased by 20%, what number do we get?

2.78 David’s swimming-pool holds 50,000 gal of water. Because of a leak, the number of gallons decreased by 12%. How many gallons of water are left in the pool?

2.79 Sales for the year 2000 were $150 million, an increase of 20% over 1999 sales. What were the 1999 sales?

2.80 The wholesale price of milk is $2.80 a gallon. What will the new price be if the farmers impose a 5% increase in the price?

2.81 How much will a $175 VCR cost if it goes on sale for “15% off.”

2.82 The price of a fax machine is $250 after a 20% reduction is taken. What was the price of the machine before the reduction?

Answers to Problems:

2.38 7:2
2.39 4:1
2.40 $7,500 and $10,500
2.41 Tom, $2,500; Robert, $1,750; Sally, $1,000
2.42 Equipment $150,000, advertising $100,000, salaries $250,000
2.43 food $6,500, housing $14,625, entertainment $1,625
2.44 $300,000, $400,000, and $500,000
2.45 x = 36
2.46 y =
2.47 45
2.48 36 gal
2.49 $505.75
2.50 $80,000 to the heart foundation and $60,000 to the cancer society
2.51 Winner, $350,000; loser, $200,000
2.52 9 teaspoons
2.53 225 red and 135 blue
2.54 (a) k = 6, (b) y = 90
2.55 (a) k = 150, (b) y = 10
2.56 $4,480
2.57 $350
2.58 40 days
2.59 (a) 1/80, (b) 1/8, (c) 5/4
2.60 (a) 18%, (b) 55%, (c) 32%
2.61 (a) 0.59, (b) 0.059, (c) 0.0059
2.62 (a) 16.2, (b) 180
2.63 (a) 60, (b) 300
2.64 $31.50
2.65 $2,008
2.66 $149
2.67 $3,020
2.68 $11,230
2.69 $250
2.70 (a) $96, (b) $7.20
2.71 $45
2.72 $32,000
2.73 7%
2.74 
2.75 63
2.76 $161
2.77 960
2.78 44,000 gal
2.79 $125 million
2.80 $2.94 per gallon
2.81 $148.75
2.82 $312.50