ANTMACK215 × WEALTHWISEGUYS
WealthWiseGuys Practical Mathematics

Chapter 2

Fractions

Parts of a whole: candy bags, pizza slices, recipes, and paychecks

Fractions describe parts of something. You use them when you split a bill, follow a recipe, or save part of every paycheck. This chapter covers what fractions mean, how to rewrite them, and how to add, subtract, multiply, and divide them.

2.1

Types of Fractions and Conversions

Learning objectives

After this section you will be able to:

  1. Name the numerator and denominator.
  2. Classify fractions as proper, improper, or mixed numbers.
  3. Convert between improper fractions and mixed numbers.

Real-life situation

What's in the candy bag?

A bag of 12 hard candies has 5 cherry, 4 lemon, and 3 lime. What fraction of the bag is each flavor?

Cherry: 5

Lemon: 4

Lime: 3

Figure 2.1 A 12-piece candy bag by flavor

The concept

A fraction has a numerator (top) that counts the parts you are talking about and a denominator (bottom) that counts the equal parts in one whole. Cherry candies are 5 of the 12 pieces, so cherry is 5/12 of the bag.

A proper fraction is less than 1 (numerator smaller than denominator). An improper fraction is 1 or more (numerator equal to or larger than denominator). A mixed number combines a whole number and a proper fraction, like 3 2/5.

To turn an improper fraction into a mixed number, divide: the quotient is the whole number and the remainder goes over the original denominator. To go back, multiply the whole number by the denominator and add the numerator.

Rolled spicy chips: 8

Cheese puffs: 6

Pretzels: 4

Popcorn: 2

Figure 2.1b A 20-bag snack variety pack (generic flavors, hypothetical)

Fraction → decimal → percent

3/8 = 0.375 = 37.5%

  1. Count 3 shaded parts out of 8 equal parts: 3/8.
  2. Divide numerator by denominator: 3 ÷ 8 = 0.375.
  3. Multiply the decimal by 100: 37.5%. The representation uses equal-size parts.

Original WealthWiseGuys diagram · Hypothetical teaching quantities.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • inventory

    The stock of goods a business holds ready to sell.

  • discount

    An amount taken off a regular price.

  • markup

    The amount added to a cost to set a selling price; the markup rate uses cost as its base.

Key terms

Numerator
The top number; how many parts are counted.
Denominator
The bottom number; how many equal parts make one whole.
Proper fraction
Numerator less than denominator; value less than 1.
Improper fraction
Numerator equal to or greater than denominator; value 1 or more.
Mixed number
A whole number plus a proper fraction.

Formulas & rules

  • Improper → mixed

    a/b = q + r/b

    The letters are just stand-ins for numbers

    • a = the top number (how many pieces you have)
    • b = the bottom number (how many pieces make one whole)
    • q = how many full wholes you can make
    • r = the pieces left over

    In plain words

    1. Divide the top by the bottom.
    2. The answer is your whole number.
    3. The leftover goes on top; the bottom stays the same.

    Try it with real numbers: 7/3 → 7 ÷ 3 = 2 with 1 left over → 2 1/3

  • Mixed → improper

    W n/d = (W × d + n)/d

    The letters are just stand-ins for numbers

    • W = the whole number in front
    • n = the top number of the fraction
    • d = the bottom number of the fraction

    In plain words

    1. Multiply the whole number by the bottom.
    2. Add the top number.
    3. Put that answer over the same bottom number.

    Try it with real numbers: 2 1/3 → 2 × 3 = 6, then 6 + 1 = 7 → 7/3

Worked examples

Example 1Flavor fractions

What fraction of the bag is cherry? Lemon?

Solution

  1. 1Cherry: 5 of 12 → 5/12.
  2. 2Lemon: 4 of 12 → 4/12, which simplifies to 1/3 (divide both by 4).

Answer: Cherry 5/12; lemon 4/12 = 1/3

Example 2The snack variety pack

In Figure 2.1b, what fraction of the 20 bags are rolled spicy chips? What fraction are not chips?

Solution

  1. 1Spicy chips: 8 of 20 → 8/20.
  2. 2GCD of 8 and 20 is 4: 8/20 = 2/5.
  3. 3Not chips: 20 − 8 = 12 bags → 12/20 = 3/5.
  4. 4Check: 2/5 + 3/5 = 5/5 = 1 whole pack. ✓

Answer: 2/5 spicy chips; 3/5 other snacks

Example 3Improper to mixed

Convert 17/5 to a mixed number.

Solution

  1. 117 ÷ 5 = 3 remainder 2.
  2. 2Whole number 3, fraction 2/5.

Answer: 3 2/5

Example 4Mixed to improper

Convert 4 3/8 to an improper fraction.

Solution

  1. 14 × 8 = 32.
  2. 232 + 3 = 35.
  3. 3Keep the denominator 8.

Answer: 35/8

Practice exercises 2.1

  1. Convert 22/7 to a mixed number.
  2. Convert 2 5/6 to an improper fraction.
  3. What fraction of the candy bag is lime, in lowest terms?
  4. Classify 9/9.
  5. Convert 13/4 to a mixed number.
  6. In the 20-bag snack pack, what fraction are pretzels or popcorn, in lowest terms?

Recap quiz 2.1

1. Which is a proper fraction?
2. 3 1/2 as an improper fraction:
3. In 5/12, the numerator is:

Section 2.1 recap

  • A fraction is part of a whole: the numerator counts parts, the denominator names their size.
  • Mixed numbers and improper fractions convert both ways using multiplication and addition.
Answer key with explanations: Section 2.1

Practice exercises

  1. 3 1/7
  2. 17/6
  3. 3/12 = 1/4
  4. Improper (it equals 1)
  5. 3 1/4
  6. 6/20 = 3/10

Recap quiz

  1. 7/8: 7 is less than 8.
  2. 7/2: 3 × 2 + 1 = 7, over 2.
  3. 5: The numerator is on top.
2.2

Equivalent Fractions, Lowest Terms, and Adding and Subtracting

Learning objectives

After this section you will be able to:

  1. Reduce fractions to lowest terms using the greatest common divisor.
  2. Find the least common denominator (LCD).
  3. Add and subtract fractions and mixed numbers.

Real-life situation

Same pizza, different cuts

Three equally sized pizzas are cut into equal slices: one into 2, one into 4, and one into 8. Eating 1 of 2, 2 of 4, or 4 of 8 slices means you ate the same amount.

1/2
2/4
4/8
Figure 2.2 Equivalent fractions: 1/2 = 2/4 = 4/8

The concept

Equivalent fractions name the same amount. Multiply or divide the numerator and denominator by the same nonzero number and the value does not change.

A fraction is in lowest terms when the only number dividing both the top and bottom is 1. Divide both by their greatest common divisor (GCD) to get there in one step.

You can only add or subtract fractions that share a denominator. Find the LCD (the smallest number both denominators divide into), raise each fraction to it, then add or subtract the numerators. With mixed numbers, borrow 1 whole from the whole number when the top fraction is too small.

Equivalent Fractions, Lowest Terms, and Adding and Subtracting: solution map

  1. 1

    Identify the question

    Reduce 24/36 to lowest terms.

  2. 2

    Calculation 1

    Divisors of 24 and 36 include 2, 3, 4, 6, 12. The GCD is 12.

  3. 3

    Calculation 2

    24 ÷ 12 = 2; 36 ÷ 12 = 3.

  4. 4

    Interpret and check

    2/3

Figure 2.2.V Equivalent Fractions, Lowest Terms, and Adding and Subtracting: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • cost

    The amount paid to make or buy an item before it is sold.

  • selling price

    The price a customer pays for an item or service.

  • revenue

    The total money a business brings in from sales, before any costs are paid.

Key terms

Equivalent fractions
Different fractions with the same value.
Greatest common divisor (GCD)
The largest number that divides two numbers evenly.
Lowest terms
A fraction whose numerator and denominator share no divisor except 1.
Least common denominator (LCD)
The smallest number every denominator divides into evenly.

Formulas & rules

  • Reduce

    a/b = (a ÷ GCD)/(b ÷ GCD)

  • Raise

    a/b = (a × k)/(b × k)

  • Like denominators

    a/d ± c/d = (a ± c)/d

Worked examples

Example 1Reduce

Reduce 24/36 to lowest terms.

Solution

  1. 1Divisors of 24 and 36 include 2, 3, 4, 6, 12. The GCD is 12.
  2. 224 ÷ 12 = 2; 36 ÷ 12 = 3.

Answer: 2/3

Example 2Add with an LCD

1/4 + 1/6

Solution

  1. 1Multiples of 6: 6, 12. 4 divides 12, so LCD = 12.
  2. 21/4 = 3/12; 1/6 = 2/12.
  3. 33/12 + 2/12 = 5/12.

Answer: 5/12

Example 3Subtract mixed numbers with borrowing

3 1/2 − 1 3/4

Solution

  1. 1LCD 4: 3 1/2 = 3 2/4.
  2. 22/4 is smaller than 3/4, so borrow 1 whole: 3 2/4 = 2 6/4.
  3. 32 6/4 − 1 3/4 = 1 3/4.

Answer: 1 3/4

Practice exercises 2.2

  1. Reduce 18/42.
  2. Raise 3/5 to a fraction with denominator 40.
  3. Find the LCD of 1/3 and 1/8.
  4. 2/3 + 1/4
  5. 5 1/6 − 2 2/3

Recap quiz 2.2

1. GCD of 16 and 40:
2. 1/2 + 1/3 =
3. Which is equivalent to 3/4?

Section 2.2 recap

  • Equivalent fractions multiply (or divide) numerator and denominator by the same number.
  • Add and subtract only when denominators match — find a common denominator first.
Answer key with explanations: Section 2.2

Practice exercises

  1. 3/7 (GCD 6)
  2. 24/40
  3. 24
  4. 8/12 + 3/12 = 11/12
  5. 31/6 − 16/6 = 15/6 = 2 1/2

Recap quiz

  1. 8: 8 divides both; 16 does not divide 40.
  2. 5/6: 3/6 + 2/6 = 5/6.
  3. 9/12: Multiply top and bottom by 3.
2.3

Multiplying and Dividing Fractions

Learning objectives

After this section you will be able to:

  1. Multiply fractions and mixed numbers, canceling when possible.
  2. Divide fractions using the reciprocal.
  3. Find a fraction of a dollar amount.

Real-life situation

Scaling a cookie recipe for a bake sale

Aaliyah's cookie recipe uses 3/4 cup of sugar per batch. She is making 2 1/2 batches for a bake sale and has 6 yards of ribbon to cut into 3/4-yard pieces for gift bags.

Brown Butter Cookies — 1 batch

  • 2 1/4 cups flour
  • 3/4 cup sugar
  • 1/2 cup brown butter
  • 1 egg
  • 1 tsp vanilla
Figure 2.3 Recipe card (one batch)

The concept

To multiply fractions, multiply the numerators and multiply the denominators. Change any mixed numbers to improper fractions first. You can cancel a factor that appears in any numerator and any denominator before multiplying.

To divide, keep the first fraction, change division to multiplication, and flip the second fraction (use its reciprocal).

The word "of" in a money problem usually means multiply: 2/5 of $450 is 2/5 × 450.

Recipe quantities in cups

Flour2.25
Sugar0.75
Brown butter0.5

Bars share a zero baseline.

Figure 2.3.V Recipe quantities in cups. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • profit

    What is left when expenses are subtracted from revenue.

  • sales tax

    A percentage added by the government to the price of many purchases.

  • break-even point

    The sales level where revenue exactly equals total cost, so there is no profit and no loss.

Key terms

Reciprocal
A fraction flipped upside down; a number times its reciprocal equals 1.
Canceling
Dividing a numerator and a denominator by a common factor before multiplying.

Formulas & rules

  • Multiply

    a/b × c/d = (a × c)/(b × d)

  • Divide

    a/b ÷ c/d = a/b × d/c

Worked examples

Example 1Scaling sugar

How much sugar for 2 1/2 batches?

Solution

  1. 12 1/2 = 5/2.
  2. 23/4 × 5/2 = 15/8.
  3. 315/8 = 1 7/8.

Answer: 1 7/8 cups

Example 2Cutting ribbon

How many 3/4-yard pieces come from 6 yards?

Solution

  1. 16 ÷ 3/4 = 6/1 × 4/3.
  2. 2Cancel 3: 2/1 × 4/1 = 8.

Answer: 8 pieces

Example 3A fraction of a paycheck

Jordan saves 2/5 of a $450 paycheck. How much is saved?

Solution

  1. 12/5 × 450 = 900/5.
  2. 2900 ÷ 5 = 180.

Answer: $180

Practice exercises 2.3

  1. 2/3 × 9/10
  2. 4 ÷ 2/3
  3. 1 1/2 × 1 1/3
  4. 7/8 ÷ 7/16
  5. 3/8 of 64 students are in band. How many?

Recap quiz 2.3

1. The reciprocal of 5/9 is:
2. 1/2 ÷ 1/4 =
3. 3/4 of $20 is:

Section 2.3 recap

  • To multiply fractions, multiply across; to divide, multiply by the reciprocal.
  • Simplify before multiplying to keep the numbers small.
Answer key with explanations: Section 2.3

Practice exercises

  1. 18/30 = 3/5
  2. 4 × 3/2 = 6
  3. 3/2 × 4/3 = 2
  4. 7/8 × 16/7 = 2
  5. 24 students

Recap quiz

  1. 9/5: Flip numerator and denominator.
  2. 2: 1/2 × 4/1 = 2.
  3. $15: 20 ÷ 4 × 3 = 15.

Chapter 2 Organizer and Review

Chapter Organizer

Every section at a glance: key formulas, key terms, and the main idea.

  • 2.1 Types of Fractions and Conversions

    Key formulas

    • Improper → mixed: a/b = q + r/b
    • Mixed → improper: W n/d = (W × d + n)/d

    Key terms

    Numerator, Denominator, Proper fraction, Improper fraction, Mixed number

    Remember

    A fraction is part of a whole: the numerator counts parts, the denominator names their size.

  • 2.2 Equivalent Fractions, Lowest Terms, and Adding and Subtracting

    Key formulas

    • Reduce: a/b = (a ÷ GCD)/(b ÷ GCD)
    • Raise: a/b = (a × k)/(b × k)
    • Like denominators: a/d ± c/d = (a ± c)/d

    Key terms

    Equivalent fractions, Greatest common divisor (GCD), Lowest terms, Least common denominator (LCD)

    Remember

    Equivalent fractions multiply (or divide) numerator and denominator by the same number.

  • 2.3 Multiplying and Dividing Fractions

    Key formulas

    • Multiply: a/b × c/d = (a × c)/(b × d)
    • Divide: a/b ÷ c/d = a/b × d/c

    Key terms

    Reciprocal, Canceling

    Remember

    To multiply fractions, multiply across; to divide, multiply by the reciprocal.

Cumulative Review

Mixed practice from this chapter and the two chapters before it.

  1. Section 2.1

    In the 20-bag snack pack, what fraction are pretzels or popcorn, in lowest terms?

    Show answer

    6/20 = 3/10

  2. Section 2.2

    5 1/6 − 2 2/3

    Show answer

    31/6 − 16/6 = 15/6 = 2 1/2

  3. Section 2.3

    3/8 of 64 students are in band. How many?

    Show answer

    24 students

  4. Review of Section 1.3

    125 × 48

    Show answer

    6,000

All prices, stores, people, and accounts in this chapter are hypothetical and for teaching only. Product references are generic categories; no brand endorses or is endorsed by this textbook.

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