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Q.3 Multiply with mixed numbers: word problems

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What does it mean to multiply mixed numbers?

Definition — Multiplying mixed numbers means finding the product when at least one factor is a mixed number (a whole number and a fraction together). To multiply correctly, change mixed numbers to improper fractions, multiply numerators and denominators, then simplify and, if needed, convert back to a mixed number.

Example:

Multiply 2 12 × 3.

  • Step 1 — Convert the mixed number to an improper fraction: 2 12 = 52 (because 2×2 + 1 = 5).
  • Step 2 — Multiply: 52 × 3 = 152.
  • Step 3 — Simplify / convert: 152 = 7 12.
Note

Always convert mixed numbers to improper fractions before multiplying. This prevents mistakes and makes multiplication straightforward.

How to multiply mixed numbers

Definition — A reliable procedure students should follow every time.

Steps:
  1. Change each mixed number to an improper fraction. (Whole × denominator + numerator = new numerator.)
  2. Multiply the numerators together to get the new numerator.
  3. Multiply the denominators together to get the new denominator.
  4. Simplify the fraction. If the numerator is larger than the denominator, convert to a mixed number.
Worked example: Multiply 1 34 × 2 25.
  • Convert: 1 34 = 74; 2 25 = 125.
  • Multiply: 74 × 125 = 8420.
  • Simplify: 8420 ÷ 4 = 215 = 4 15.
Note

Look for opportunities to simplify (cancel common factors) before multiplying to make the calculation easier and keep numbers smaller.

Strategies for word problems

Definition — A plan for decoding the words, identifying what to multiply, and showing work clearly.

Helpful steps:
  • Read the problem twice and underline important numbers and units.
  • Decide whether to multiply a mixed number by a whole number, a fraction, or another mixed number.
  • Write an equation, convert mixed numbers to improper fractions, then multiply.
  • Label the answer with correct units and check whether the result is reasonable.
Example one-step problem:

Rosa bakes 3 12 batches of cookies. Each batch uses 1 23 cups of flour. How many cups of flour does she use?

  • Equation: 3 12 × 1 23.
  • Convert: 3 12 = 72; 1 23 = 53.
  • Multiply: 72 × 53 = 356 = 5 56 cups.
Note

Always include units (cups, yards, hours). Units help you decide whether your answer is reasonable.

One-step word problems — common combinations

Definition — One-step problems ask for a single multiplication operation. Here are common pairs (use two at a time): mixed × whole, whole × fraction, fraction × mixed, mixed × mixed.

Examples (one for each combination):
  • Mixed × Whole: 2 14 boxes × 6 pencils per box = ?
    Work: 2 14 = 94; 94×6 = 544=13 24=13 12 pencils.
  • Whole × Fraction: 4 hours × 34 of an hour per lesson = ?
    Work: 4 × 34 = 124 = 3 lessons.
  • Fraction × Mixed: 25 of a yard × 1 12 yards per piece = ?
    Work: convert 1 12 = 32; multiply 25 × 32 = 610 = 35 yards.
  • Mixed × Mixed: 1 23 × 2 14 = ?
    Work: 1 23 = 53; 2 14 = 94; multiply = 4512 = 3 912 = 3 34.
Note

When a whole number is present, think of it as a fraction with denominator 1 (for example, 6 = 61). That helps keep the multiplication method consistent.

Two-step word problems

Definition — Two-step problems combine multiplication with another operation (addition or subtraction). Read carefully to decide which operation comes first.

Examples:
  • Carla walks 1 14 miles each day for 3 days, then she walks 1 more mile on the fourth day. How many miles total?
    Work: 1 14 × 3 = 54 × 3 = 154 = 3 34. Add the extra 1: 3 34 + 1 = 4 34.
  • Sam buys 2 boxes of tiles. Each box covers 2 12 square feet. He needs an extra 34 square foot. How many square feet will he have after using both boxes and adding the extra piece?
    Work: 2 × 2 12 = 2 × 52 = 102 = 5. Add extra 34 = 5 + 34 = 5 34.
Note

Underline action words (total, each, left, extra) to decide whether to multiply first or add/subtract first. Show units with answers.

All combinations — quick reference

Definition — This section lists every two-number combination students should be comfortable multiplying.

Combinations and short guides:
  • Whole × Whole — multiply as usual (treat as fractions with denominator 1 if helpful).
  • Whole × Fraction — multiply whole × numerator, divide by denominator (or convert whole to fraction n1 and multiply).
  • Whole × Mixed — convert mixed to improper, then multiply.
  • Fraction × Fraction — multiply numerators and denominators; simplify.
  • Fraction × Mixed — convert mixed to improper, then multiply.
  • Mixed × Mixed — convert both to improper fractions, multiply, simplify, convert back if needed.
Short worked example showing every pair (two-number pairs):
  1. Whole × Fraction: 3 × 25 = 65 = 1 15.
  2. Whole × Mixed: 4 × 1 13 = 4 × 43 = 163 = 5 13.
  3. Fraction × Fraction: 34 × 12 = 38.
  4. Fraction × Mixed: 23 × 2 14 = 23 × 94 = 1812 = 32 = 1 12.
  5. Mixed × Mixed: 1 12 × 1 23 = 32 × 53 = 156 = 2 36 = 2 12.
Note

Memorize the procedure (convert → multiply → simplify → convert back) and practice every pair so you can pick the right method quickly during tests.

Practice problems (use these like quick checks)

Definition — Short problems to practice the skills shown above. Try all combinations and check your reasoning.

Problems:
  1. Multiply: 2 13 × 3.
  2. Multiply: 4 × 25.
  3. One-step word problem: A recipe needs 1 34 cups of sugar for one batch. How much sugar for 5 batches?
  4. Two-step word problem: Kayla runs 1 12 miles each day for 4 days, then 2 34 miles on the fifth day. How many miles total?
  5. Mixed × Mixed: 2 25 × 1 310.
Note

Try the problems on scratch paper, show all steps (conversion, multiplication, simplification), and check units. If you get stuck, go back to the step-by-step section.

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