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Subtraction can be denoted by

less than

subtracted from

decreased byexceeds

shorter than

difference between

Multiplication can be denoted by

product

times

multiplied by

twice as large

three times a number12

of a number

Division can be denoted by

divided by

quotient of

Equals can be denoted by

is

will be

is equal to

Here are some examples of how these phrases are translated into symbols:

Word Statement Symbolic Representation

Five times a number x 5x

Three more than a number x x 3

A number x decreased by 7 x 7

One-half of a number x 1

2

x or x

2

The square of a number x2

Nine added to twice a number x 9 2x

Four times a number minus eight 4x 8

The cost of x feet of rope at 15 cents a foot 0.15x

LESSON 8 Algebraic Representation 83

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Try These

Write each in symbols:

1. Six times a number x decreased by four.

2. A number x increased by six.

3. Eight less than a number x.

4. Fifteen minus one-third of a number x.

5. The square of a number x plus 7.

SOLUTIONS:

1. 6x 4

2. x 6

3. x 8

4. 15 13

x or 15 x3

5. x2 7

In the previous examples, only one unknown was used. At other times, it is

necessary to represent two relatedunknowns by using one variable. Consider

these examples:

The sum of two numbers is 12. When you are given two numbers whose

sum is 12 and one number is, say, 7, how would you find the other number?

You would say 12 7. So if one number is x, the other number would be12 x.

One number is six more than another number. If I told you one number is

10, how would you find the other number? You would add 10 6. So if one

stated number is x, the other number would be x 6.

One number is five times another number. If I told you one number is

two, how would you find the other number? You would multiply two by five.

So if one number is x, the other number would be 5x.

One number is twelve less than another number. If I told you one number

was 30, how would you find the other number? You would subtract 30 12.

So if one number is x, the other number is x 12.

Try These

Represent each using symbols:

1. The sum of two numbers is 15.

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2. One number is four times another number.

3. One number is 6 more than twice the other number.

4. One number is 4 less than one-half another number.

5. The difference of two numbers is 12.

SOLUTIONS:

1. Let x the first number and 15x the second number.

2. Let x the first number and 4x the second number.

3. Let x the first number and 2x 6 the second number.

4. Let x the first number and 12x 4 the second number.

5. Let x the first number and x 12 the second number.

The third skill necessary to solve word problems is to be able to translate

the given symbols of representation into an equation. Consider theseexamples.

Three times a number increased by 10 is equal to 28 translates to

3 x 10 28

Seven times a number decreased by 6 is equal to 29.

7 x 6 29

The difference between a number and one-half itself is equal to 12.

x 12x 12

Try These

Translate each into an equation:

1. Seven increased by four times a number is 31.

2. If 5 is subtracted from two times a number, the answer is 15.

3. One half a number plus 8 is equal to 19.

4. Two times a number plus $1.20 is equal to $1.80.5. The sum of a number and two times itself is equal to 30.

LESSON 8 Algebraic Representation 85

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SOLUTIONS:

1. 7 4x 31

2. 2x 5 15

3. 12x 8 194. 2x 1.20 1.80

5. x 2x 30

Finally, it is necessary to be able to write an equation for two related

unknowns using one variable.

EXAMPLE: Write an equation for this problem: One number is 10 more

than another number and their sum is 16.

SOLUTION: Let x the smaller number and x 10 the larger number.

The equation is xx 10 16.

EXAMPLE: Write an equation for this problem: One number is five times

as large as another number. If three times the smaller number is subtracted

from the larger number, the answer is 20.

SOLUTION: Let x the smaller number and 5x the larger number. The

equation is 5x 3x 20.

Try TheseWrite an equation for each:

1. The smaller number is 12

of the larger number. Find the numbers if

their sum is 36.

2. A certain number exceeds another number by 6. If their sum is 56, find

the numbers.

3. One number is 8 more than twice another number. Find the numbers if

their sum is 50.4. What number increased by 1

4of itself is equal to 5?

5. Two times a number is 6 more than 12

the number. Find the numbers.

LESSON 8 Algebraic Representation86