Questions in introductory algebra

1. Solve the equation x + 5 = 19

2. Find two numbers that add to 6 and multiply to 8.

3. Sam paid his bill of $9.85 with only quarters, dimes

and nickels.

He used twice as many quarters as nickels, and more nickels

than dimes.

How many quarters, dimes, nickels did he use?

4. Sam paid his bill of $9.85 with only quarters, dimes and

nickels.

He used twice as many quarters as nickels, and one less

nickel than dimes.

How many quarters, dimes, nickels did he use?

5. Sam paid his bill of $9.85 with only quarters, dimes and

nickels.

He used twice as many quarters as nickels, and one less

quarter than dimes.

How many quarters, dimes, nickels did he use?

Question # 1

Solve the equation x + 5 = 19

The equation x + 5 = 19 means

there is a number that we name x,

and when we add 5 to that number, we get 19.

The solution is to subtract 5 from 19.

( x + 5) - 5 = 19 - 5

x + 5 - 5 = 14

x = 14

Verify that the answer is correct.

14 + 5 = 19

19 = 19

Go to Top

Main Menu

Question #2

Find two numbers that add to 6 and multiply to 8.

It's easy to think of the answer to this, without using

algebra.

But since this is an algebra question, let's use algebra to

answer the question.

Let x and y be the names of the two numbers.

Then

x + y = 6

x * y = 8

solve the first equation for y.

y = 6 - x

Plug into the second equation.

x * (6 -x) = 8

6 * x - x**2 = 8

Move all the terms to one side of the equation,

changing signs of terms that cross the equal sign.

x**2 - 6 * x + 8 = 0

This is a quadratic equation that can be solved, either by

factoring, or by using the quadratic formula.

Either method gives solutions x = 2 or x = 4.

So take one of 2 or 4 to be x, and the other to be y.

The two numbers are 2 and 4.

Verify that the answer is correct.

2 + 4 = 6

2 * 4 = 8

Go to Top

Main Menu

Question # 3

Sam paid his bill of $9.85 with only quarters, dimes

and nickels.

He used twice as many quarters as nickels, and more nickels

than dimes.

How many quarters did he use?

How many dimes did he use?

How many nickels did he use?

Let q be the name of the number of quarters.

Let d be the name of the number of dimes.

Let n be the name of the number of nickels.

25 q + 10 d + 5 n = 985

q = 2 n

n > d

25 ( 2 n) + 10 d + 5 n = 985

50 n + 10 d + 5 n = 985

55 n + 10 d = 985

11 n + 2 d = 197

197 has a remainder of 10 when divided by 11

11 * 17 + 10 = 197

11 * 17 + 2 * 5 = 197

11 n + 2 d = 197

n > d

n could be 17 and d could be 2 since 17 > 2.

Could n be 16?

if n is 16, 11 * 16 + 2 d = 197

2 d = 197 - 11 * 16 = 197 - 176 = 21.

n cannot be 16 because 21 is odd.

could n be 15?

if n is 15, 11 * 15 + 2 d = 197

2 d = 197 - 11 * 15 = 197 - 165 = 32

d = 16.

n cannot be 15 because then d would be 16, which is > 15.

n = 17, 2 d = 197 - 11*17 = 197 - 187 = 10

d = 5

q = 2*n = 2 * 17 = 34

Check answer.

34 * 25 + 5 * 10 + 17 * 5

= 850 + 50 + 85 = 985

Go to Top

Main Menu

Question # 4. Sam paid his bill of $9.85 with only quarters, dimes and

nickels.

He used twice as many quarters as nickels, and one less

nickel than dimes.

How many quarters did he use?

How many dimes did he use?

How many nickels did he use?

25 q + 10 d + 5 n = 985

q = 2 * n

n = d - 1

d = n + 1

25 * (2 * n) + 10 * (n + 1) + 5 n = 985

50 * n + 10 * n + 10 + 5 * n = 985

(50 + 10 + 5) * n + 10 = 985

65 * n = 975

13 * n = 195

n = 15

d = n + 1 = 16

q = 2 * n = 30

Check answer.

30 * 25 + 16 * 10 + 15 * 5 = 750 + 160 + 75 = 985

Go to Top

Main Menu

Question # 5.

Sam paid his bill of $9.85 with only quarters, dimes and

nickels.

He used twice as many quarters as nickels, and one less

quarter than

dimes.

How many quarters did he use?

How many dimes did he use?

How many nickels did he use?

25 q + 10 d + 5 n = 985

q = 2 * n

d = q + 1 = 2 * n + 1

25 * (2 * n) + 10 * ( 2 * n + 1) + 5 * n = 985

50 * n + 20 * n + 10 + 5 * n = 985

(50 + 20 + 5) * n = 985 - 10 = 975

75 * n = 975

15 * n = 195

3 * n = 39

n = 13

q = 2 * n = 26

d = q + 1 = 27

Test correctness of answer.

26 * 25 + 27 * 10 + 13 * 5 = 650 + 270 + 65 = 985

Go to Top

Main Menu