Business Mathematics: Review of Arithmetic - Basic Algebra

In algebra, symbols, called variables, are used to represent numbers whose values are unknown. This enables us to work with them even though we don’t yet know what their values are.

Example
If p represents the price, in dollars, of a commodity, and we wish to increase its price by 3 dollars, its new price is represented as p + 3.

Example
The cost of manufacturing a computer is c dollars. What is the cost of manufacturing 10 computers?
We multiply the cost of one computer by 10. Hence the cost becomes 10c. Note that 10c is the same as 10 × c. In algebra, if a symbol is omitted, multiplication is assumed.

Equations are used to represent equality between two quantities. If the cost c of an item is the same as its selling price p, we write c = p to represent that fact.
The phrase more than denotes addition, less than indicates subtraction, and the word is usually denotes equality.

Example
Express the sentence “3 more than p is 7 less than c” as an equation.
We translate phrase by phrase. The word is becomes the center of the equation and is represented by = 3 more than p becomes p + 3 and 7 less than c becomes c – 7. The resulting equation is
p + 3 = c – 7

Example
Express “5 more than a is 3 more than b” as an equation:
a + 5 = b + 3
If an equation involves an unknown variable, we can determine the value of that variable by using the following simple principle:
If you perform any arithmetic operation to one side of an equation, and perform the same operation to the other side, the equation remains valid.
By performing the right combination of operations, the values of unknown variables can be determined.

Example
Solve for x: 2x + 3 = 13.
To get to x, we first subtract 3 from both sides of the equation:

Next we divide both sides of the equation by 2:

The value of the unknown variable x is 5.

Example
Ryan orders three TV sets and leaves a $50 deposit. If the balance due is $1,225, how much is each television set?
We represent the cost of a television set by x. The three sets will cost 3x dollars. Since he left a $50 deposit, the balance remaining on the purchase is 3x – 50 dollars. Thus 3x – 50 = 1,225.

Each set costs $425.

 

Solved Problems

1.35   If a refrigerator costs × dollars and goes on sale for $50 off, how much will it cost to purchase?
Solution
x – 50 dollars.

1.36   Express the sentence “12 more than x is 5 less than _y” as an equation.
Solution
x + 12 = y–5.

1.37   Solve for x: 5x – 7 = 33.
Solution


1.38   Solve for y: 2y/3 = 16.
Solution


1.39   Solve for b: b/14 = 3/7.
Solution
Cross–multiply to get 7b = 42. Then divide both sides by 7. b = 6.

1.40   Solve for w: (2w – 3)/9 = 3.
Solution


1.41   Bill orders five VCRs and two TV sets for a total cost of $1,260. Each television set costs $390. How much is one VCR?
Solution
Let v represent the cost of one VCR. Since two TV sets cost $780 (2 × 390 = 780), we have the equation 5v + 780 = 1,260. Now solve for v:


Supplementary Problems

1.42 How many hundreds, tens, and ones are represented by the number 659?
1.43 How many millions, hundred thousands, ten thousands, thousands, hundreds, tens, and ones are represented by the number 7,684,713?
1.44 Add the numbers 579 and 317.
1.45 Find the sum of 1,582, 2,359, and 3,456.
1.46 Subtract 782 from 953.
1.47 Add 5,392 to 7,683 and then subtract 8,567.
1.48 Multiply 29 by 57.
1.49 Find the product of 173 and 62.
1.50 Divide 3,526 by 43.
1.51 Find the quotient of 1,170 and 45.
1.52 Find the remainder if 257 is divided by 19.
1.53 Which of the following numbers is not prime: 13, 17, 23, 37, 57?
1.54Find the prime factorization of 2,520.
1.55 Express the fraction 3/4 three different ways.
1.56 Reduce each of the following fractions to lowest terms:
1.57 Which of the following fractions is not equivalent to 2/3?
1.58 Combine as a single fraction and then simplify:
1.59 Combine as a single fraction and then simplify:
1.60 Express the fraction 7/9 as an equivalent fraction whose denominator is 72.
1.61 Find the LCD of the fractions 5/21 and 7/35. Then add the fractions together.
1.62 Find the LCD of the fractions 5/99 and 1/24. Then subtract the second from the first.
1.63 Combine as a single fraction: 2/3 + 3/4 – 1/8.
1.64 Combine as a single fraction: 3/10 + 2/15 – 1/35.
1.65 Multiply 5/7 by 14/35 and reduce.
1.66 Multiply 3/5, 2/9, and 5/8. Be sure to reduce.
1.67 Convert to a fraction.
1.68 Convert 72/11 to a mixed number.
1.69 Evaluate (a) 3 + 4·6 (b) (3+ 4)·6
1.70 Evaluate (a) 5 + 32·6 (b) (5 + 3)2·6
1.71 Evaluate (a) b)
1.72 Add: 2.76 + 3.52 + 6.79.
1.73 Add: 3.1+5.12 + 2.73.
1.74 Subtract 5.311 from 7.512.
1.75Subtract 2.753 from 5.611.
1.76 Multiply 3.41 by 2.123.
1.77 Multiply 6.21 by the sum of 1.23 and 2.34.
1.78 Add 6.21 to the product of 1.23 and 2.34.
1.79 Divide 164.7 by 13.5.
1.80 Divide 14.875 by 1.25.
1.81 Convert to a fraction: (a) 0.75, (b) 0.625, (c) 0.45.
1.82 Convert to a decimal: (a) 33/100, (b) 7/8, (c) 11/32.
1.83 Use a calculator to compute the answers to the following problems: (a) 17.23 + 5.367, (b) 19.1 –6.57, (c) 19.6 × 11.25,(d) 15.2 + 12.6 – 11.1, (e) 12.75 ÷25, (f) 5.23.
1.84 Use a calculator to compute the answers to the following problems: (a) 1.25 + 2.67×3.47, (b) (1.25 + 2.67) × 3.47, (c) 7.5 ÷ 1.25 × 3.4, (d) 7.5 ÷ (1.25 × 3.4).
1.85 Compute each of the following using your calculator (round to four decimal places):
1.86 If p represents the price (in dollars) of a new car and the manufacturer gives a $2,000 rebate, express the actual cost of the car in terms of p.
1.87 If an apple costs a cents and a banana costs b cents, express the cost of five apples and seven bananas in terms of a and b.
1.88 Express the sentence "5 more than × is 9 less than twice y" as an equation.
1.89 Solve for x: 3x + 5 = 38.
1.90 Solve for x: (JC/15) = 1/5.
1.91 Solve for x: (2x – 3)/7 = 5.
1.92 Peter buys three tacos from Taco City. He gives the waiter $5.00 and gets $2.15 change. How much is one taco?
1.93 Walter orders eight chairs for his dining-roomtable. He leaves a $75 deposit. If the balance due is $845, how much is each chair?
1.94  Samuel goes to the store and buys five videotapes and seven rolls of film. He pays $30.95. If each videotape costs $3.25, how much is a roll of film?

Answers to Supplementary Problems

1.42 6 hundreds, 5 tens, and 9 ones
1.43 7 millions, 6 hundred thousands, 8 ten thousands, 4 thousands, 7 hundreds, 1 ten, and 3 ones
1.44 896
1.45 7,397
1.46 171
1.47 4,508
1.48 1,653
1.49 10,726
1.50 82
1.51 26
1.52 10
1.53 57 is not prime; 57 = 19 × 3
1.54 22 · 32 · 5 · 7
1.55 6/8, 9/12, and 15/20
1.56 (a) 3/4, (b) 2/3, (c) 13/17
1.57 38/59
1.58 2/9
1.59 1/2
1.60 56/72
1.61 The LCD is 105; 46/105
1.62 The LCD is 792; 73/792
1.63 31/24
1.64 85/210 = 17/42
1.65 2/7
1.66 1/12
1.67 37/7
1.68 
1.69  (a) 27, (b) 42
1.70  (a) 59, (b) 384
1.71 (a) 29/30, (b) 3/10
1.72 13.07
1.73 10.95
1.74 2.201
1.75 2.858
1.76 7.23943
1.77 22.1697
1.78 9.0882
1.79 12.2
1.80 11.9
1.81 (a) 3/4, (b) 5/8, (c) 9/20
1.82 (a) 0.33, (b) 0.875, (c) 0.34375
1.83 (a) 22.597, (b) 12.53, (c) 220.5, (d) 16.7, (e) 10.2, (/) 140.608
1.84 (a) 10.5149, (b) 13.6024, (c) 20.4, (d) 1.764705882
1.85 (a) 2,829.4175, (b) 1,372.8162, (c) 417.6330
1.86 p – 2,000 dollars
1.87 5a + lb
1.88 x + 5 = 2y – 9
1.89 x = ll
1.90 × = 3
1.91 × = 19
1.92 $0.95
1.93 $115
1.94 $2.10