Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview
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Chapter 3
Worked example (no skipped steps)
3.1 Simplify 12 .
18
The greatest common factor of 12 and 18 is 6. Divide the numerator and denominator by 6: 12
12 ÷ 6
2
=
=
18
18 ÷ 6
3
Answer: 2 .
3
Worked example (no skipped steps)
3.2 Compute 1 + 3 .
4
8
The denominators are 4 and 8. A denominator of 8 works because 8 is a multiple of both 4
and 8.
Rewrite 1 in eighths:
4
1
1 · 2
2
=
=
4
4 · 2
8
141
Now both fractions have the same denominator:
2
3
5
+
=
8
8
8
Answer: 5 .
8
Worked example (no skipped steps)
3.3 Compute 5 − 1 .
6
3
Use denominator 6. Rewrite 1 as sixths:
3
1
1 · 2
2
=
=
3
3 · 2
6
Now subtract:
5
2
3
−
=
6
6
6
Simplify by dividing top and bottom by 3:
3
3 ÷ 3
1
=
=
6
6 ÷ 3
2
Answer: 1 .
2
Worked example (no skipped steps)
3.4 Compute 2 · 9 .
3
4
Multiply numerators and denominators:
2
9
2 · 9
18
·
=
=
3
4
3 · 4
12
Simplify by dividing top and bottom by 6:
18
18 ÷ 6
3
=
=
12
12 ÷ 6
2
Answer: 3 .
2
Worked example (no skipped steps)
3.5 Compute 3 ÷ 9 .
5
10
Division of fractions means multiply by the reciprocal:
3
9
3
10
÷
=
·
5
10
5
9
Now multiply:
3
10
3 · 10
30
·
=
=
5
9
5 · 9
45
Step-by-step solutions
Simplify by dividing top and bottom by 15:
30
30 ÷ 15
2
=
=
45
45 ÷ 15
3
Answer: 2 .
3
Worked example (no skipped steps)
3.6 Simplify 15 .
35
The greatest common factor is 5. So:
15
15 ÷ 5
3
=
=
35
35 ÷ 5
7
Answer: 3 .
7
Worked example (no skipped steps)
3.7 Compute 7 + 1 .
8
8
The fractions already have the same denominator:
7
1
8
+
=
8
8
8
Since 8 = 1,
8
7
1
+
= 1
8
8
Answer: 1.
Worked example (no skipped steps)
3.8 Compute 5 · 6 .
12
3
Multiply:
5
6
5 · 6
30
·
=
=
12
3
12 · 3
36
Simplify by dividing top and bottom by 6:
30
30 ÷ 6
5
=
=
36
36 ÷ 6
6
Answer: 5 .
6
143
Worked example (no skipped steps)
3.9 Compute 2 + 1 .
5
10
Use denominator 10. Rewrite 2 as tenths:
5
2
2 · 2
4
=
=
5
5 · 2
10
Now add:
4
1
5
+
=
10
10
10
Simplify by dividing top and bottom by 5:
5
1
=
10
2
Answer: 1 .
2
Worked example (no skipped steps)
3.10 Compute 7 − 3 .
8
4
Use denominator 8. Rewrite 3 as eighths:
4
3
3 · 2
6
=
=
4
4 · 2
8
Now subtract:
7
6
1
−
=
8
8
8
Answer: 1 .
8
Worked example (no skipped steps)
3.11 Compute 4 · 3 .
9
2
Multiply:
4
3
4 · 3
12
·
=
=
9
2
9 · 2
18
Simplify by dividing top and bottom by 6:
12
2
=
18
3
Answer: 2 .
3
Step-by-step solutions
Worked example (no skipped steps)
3.12 Compute 5 ÷ 10 .
6
9
Flip the second fraction and multiply:
5
10
5
9
÷
=
·
6
9
6
10
Now multiply:
5
9
45
·
=
6
10
60
Simplify by dividing top and bottom by 15:
45
45 ÷ 15
3
=
=
60
60 ÷ 15
4
Answer: 3 .
4
Worked example (no skipped steps)
3.13 Simplify 18 .
24
The greatest common factor is 6. Divide top and bottom by 6: 18
18 ÷ 6
3
=
=
24
24 ÷ 6
4
Answer: 3 .
4
Worked example (no skipped steps)
3.14 Compute 3 + 2 .
7
7
Same denominator:
3
2
5
+
=
7
7
7
Answer: 5 .
7
Worked example (no skipped steps)
3.15 Compute 11 − 1 .
12
3
Use denominator 12. Rewrite 1 as twelfths:
3
1
1 · 4
4
=
=
3
3 · 4
12
Now subtract:
11
4
7
−
=
12
12
12
Answer: 7 .
12
145
Worked example (no skipped steps)
3.16 Compute 2 1 + 3 .
4
4
Rewrite the mixed number:
1
1
2
= 2 +
4
4
Now add the fractions:
1
3
4
+
=
= 1
4
4
4
So:
1
3
2 +
+
= 2 + 1 = 3
4
4
Answer: 3.