Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview
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Chapter goals
By the end of this chapter, you will be able to:
• simplify fractions correctly,
• add and subtract fractions with a common denominator,
• multiply and divide fractions without confusion,
• use mixed numbers in a calm, controlled way,
• understand why fraction control matters in later algebra.
Prerequisites (if this is hard, go back to Unit 0)
You need Unit 0 fraction language: numerator, denominator, equivalent fractions, and denominator not zero.
Why this matters
Key idea
Fractions are where many students stop trusting themselves. Then every later chapter feels harder than it should.
But fractions are mechanical once the rules are clear.
Real life: what it’s for
Fractions are everywhere: recipes, medicine doses, scale drawings, probability, tax rates, discounts, and averages. In algebra, fractions also appear inside equations, formulas, slopes, derivatives, and standard deviations.
37
Chapter 3. Fractions without panic
Equivalent fractions
Key idea
This chapter assumes you already saw the full logic in Unit 0. Here is the short version: if you multiply or divide the numerator and denominator by the same nonzero number, you change the look of the fraction, not its value. That is why simplification works.
Prerequisites (if this is hard, go back to Unit 0)
If the phrase “same nonzero number” still feels vague, go back to Unit 0 before pushing forward.
Fractions become manageable only when that move feels legal and normal.
Worked example (no skipped steps)
Simplify:
12
18
Step 1: find the greatest common factor of 12 and 18. That factor is 6.
Step 2: divide the numerator and denominator by 6.
12
12 ÷ 6
2
=
=
18
18 ÷ 6
3
Answer: 2 .
3
Adding and subtracting fractions
Recipe (steps that always work)
To add or subtract fractions:
1. find a common denominator,
2. rewrite each fraction using that denominator,
3. combine the numerators,
4. keep the denominator,
5. simplify if possible.
Adding and subtracting fractions
39
Worked example (no skipped steps)
Compute:
1
3
+
4
8
Step 1: use a common denominator of 8. That is the same move explained carefully in Unit 0: the denominator 4 can be turned into 8 by multiplying by 2, while the second fraction is already in eighths.
Step 2: rewrite 1 as 2 . Using the Unit 0 rule for equivalent fractions, 4
8
1
1 · 2
2
=
=
.
4
4 · 2
8
Step 3: add because both fractions are now written in eighths.
2
3
2 + 3
5
+
=
=
8
8
8
8
Answer: 5 .
8
Worked example (no skipped steps)
Compute:
5
1
−
6
3
Step 1: find a common denominator. A convenient choice is 6 because 6 is already the first denominator and 3 can be turned into 6 by multiplying by 2.
Step 2: rewrite the second fraction in sixths.
1
1 · 2
2
=
=
3
3 · 2
6
The first fraction stays the same:
5
5
=
6
6
Step 3: subtract the numerators.
5
2
3
−
=
6
6
6
Step 4: simplify. Both 3 and 6 are divisible by 3: 3
3 ÷ 3
1
=
=
6
6 ÷ 3
2
Answer: 1 .
2
Chapter 3. Fractions without panic
Worked example (no skipped steps)
Compute:
1
3
2
+
4
4
Step 1: keep the whole-number part separate for a moment.
1
3
1
3
2
+
= 2 +
+
4
4
4
4
Step 2: add the fractional parts. The denominators already match, so: 1
3
4
+
=
= 1
4
4
4
Step 3: add the whole numbers.
2 + 1 = 3
Answer: 3.
Multiplying and dividing fractions
Key idea
The full reason behind these rules was introduced in Unit 0. Here we focus on fluency, because these moves will soon appear inside equations, formulas, and functions.
Multiplication is easy:
𝑎
𝑐
𝑎𝑐
·
=
𝑏
𝑑
𝑏𝑑
as long as the denominators are not zero.
Division means multiply by the reciprocal:
𝑎
𝑐
𝑎
𝑑
÷
=
·
𝑏
𝑑
𝑏
𝑐
Here 𝑏, 𝑐, and 𝑑 must be nonzero: 𝑏 and 𝑑 are denominators, and 𝑐 becomes a denominator when we flip 𝑐 .
𝑑
Worked example (no skipped steps)
Compute:
2
9
·
3
4
Step 1: multiply the numerators.
2 · 9 = 18
Step 2: multiply the denominators.
3 · 4 = 12
So:
2
9
18
·
=
3
4
12
Multiplying and dividing fractions
41
Step 3: simplify. Both 18 and 12 are divisible by 6, so divide top and bottom by 6: 18
18 ÷ 6
3
=
=
12
12 ÷ 6
2
Answer: 3 .
2
Worked example (no skipped steps)
Compute:
3
9
÷
5
10
Step 1: keep the first fraction.
3
5
Step 2: flip the second one. This is the reciprocal move from Unit 0.
9
10
→
10
9
Step 3: multiply.
3
10
30
·
=
5
9
45
Step 4: simplify using the Unit 0 rule. Both 30 and 45 are divisible by 15: 30
30 ÷ 15
2
=
=
45
45 ÷ 15
3
Answer: 2 .
3
Bridge (why this matters later)
Fraction control matters before functions, trigonometry, calculus, and statistics because many formulas involve ratios. If fractions feel scary, later formulas feel scary too.
Common mistakes (and how to avoid them)
• adding denominators when adding fractions,
• forgetting to simplify,
• flipping the wrong fraction when dividing,
• multiplying only the numerator or only the denominator,
• treating a mixed number like a mysterious new object instead of a whole number plus a fraction.
Chapter 3. Fractions without panic
Recap: what you’re taking away
Fractions stop being scary when you remember three things:
• addition needs a common denominator,
• division means multiply by the reciprocal,
• mixed numbers are just whole numbers plus fractions.
That small toolkit supports a surprising amount of later math.
Problems
3.1. Simplify: 12 .
18
3.2. Compute: 1 + 3 .
4
8
3.3. Compute: 5 − 1 .
6
3
3.4. Compute: 2 · 9 .
3
4
3.5. Compute: 3 ÷ 9 .
5
10
3.6. Simplify: 15 .
35
3.7. Compute: 7 + 1 .
8
8
3.8. Compute: 5 · 6 .
12
3
3.9. Compute: 2 + 1 .
5
10
3.10. Compute: 7 − 3 .
8
4
3.11. Compute: 4 · 3 .
9
2
3.12. Compute: 5 ÷ 10 .
6
9
3.13. Simplify: 18 .
24
3.14. Compute: 3 + 2 .
7
7
3.15. Compute: 11 − 1 .
12
3
3.16. Compute: 2 1 + 3 .
4
4
Quick checklist (if you miss it, reread and try again)
• I can simplify a fraction by dividing top and bottom by the same factor.
• I can add fractions using a common denominator.
• I can divide fractions by multiplying by the reciprocal.
• I can read a mixed number as a whole number plus a fraction.
Decimals, percentages, and translating
between them