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Multiplicative Comparison Word Problems | 4.OA.A.2

Solving Word Problems With Multiplicative Comparison

From Interpreting Equations to Finding the Missing Piece

In this lesson:

  • Find a missing quantity in a multiplicative comparison — by multiplying or dividing
  • Write an equation with a symbol for the unknown before solving
  • Distinguish multiplicative comparison from additive comparison
Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Learning Objectives for Today's Lesson

By the end of this lesson, you should be able to:

  1. Identify which quantity is unknown
  2. Write an equation with a symbol
  3. Solve for any unknown position
  4. Distinguish multiplicative from additive comparison
  5. Check that your answer is reasonable
Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Quick Recall: Name the Three Parts

In the equation 12 = 4 × 3:

  • Which is the large quantity? Which is the comparison factor? Which is the reference?

Today, one of these three numbers will be missing — your job is to find it.

Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Finding a Missing Total by Multiplying

"A sunflower is 6 times as tall as a tulip. The tulip is 9 inches tall. How tall is the sunflower?"

Tape diagram: one 9-inch reference bar for the tulip, 6 copies joined to form the sunflower's bar, total marked with a question mark before being resolved to 54

Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Second Example: Try It Yourself First

"A lion weighs 8 times as much as a fox. The fox weighs 12 pounds. How much does the lion weigh?"

n = 8 × 12 = 96 pounds

Check: is 96 much more than 12? Yes — reasonable for "8 times as much."

Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Check-In: The Rope and Ribbon Problem

"A rope is 4 times as long as a ribbon. The ribbon is 7 feet long. How long is the rope?"

Write the equation with a symbol, then solve.

Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Practice: Two Unknown-Product Word Problems

Write the equation and solve for each:

  1. A building is 5 times as tall as a house (house = 14 ft)
  2. A truck carries 3 times as much as a car (car = 9 boxes)
Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Practice: Two More Unknown-Product Problems

Write the equation and solve for each:

  1. A river is 9 times as long as a stream (stream = 6 miles)
  2. A whale swims 7 times as far as a diver (diver = 8 miles)

Item 3's bigger factor makes an additive slip easiest to catch here.

Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

What If the Total Is Already Known?

You've found the total by multiplying. What if the total is known instead, and something else is missing?

One fact family: 28 = 4 × 7 ⇔ 28 ÷ 4 = 7 ⇔ 28 ÷ 7 = 4

When a piece is missing, divide the total.

Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Unknown Factor: How Many Times As Long?

"A whale is 36 feet long. A dolphin is 4 feet long. The whale is how many times as long as the dolphin?"

Tape diagram: dolphin's 4-foot reference bar shown, an unknown number of copies joined to reach the whale's known 36-foot total, with the count of copies marked with a question mark

Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Unknown Reference: How Big Is Each Share?

"Emma has 5 times as many stickers as Liam. Emma has 45 stickers. How many stickers does Liam have?"

Tape diagram: Emma's 45-sticker total bar divided into 5 equal parts, each part marked with a question mark, representing Liam's unknown share

Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Two Kinds of Division, One Operation

  • Measurement division (whale/dolphin): you know each group's size, and count how many groups fit
  • Partitive division (Emma/Liam): you know how many groups there are, and find each group's size

Three small side-by-side tape diagrams sharing the same A equals B times C structure, each with a different single part marked with a question mark: total unknown, number-of-copies unknown, size-of-each-copy unknown

Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Watch Out: Does Your Answer Make Sense?

"Maria has 5 times as many stickers as Jake. Maria has 35 stickers. How many does Jake have?"

Before you compute — predict: should Jake have more stickers than Maria, or fewer?

35 = 5 × n, n = 35 ÷ 5 = 7

Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Practice: Mixed Unknown Positions, Part 1

Identify what's unknown, then solve:

  1. A bin holds 6 times as many apples as an 8-apple basket. Find the bin.
  2. A car goes 54 miles; a bike goes 6 miles. How many times farther?
  3. A store's 63 pens are 7 times a box's amount. Find the box.
Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Practice: Mixed Unknown Positions, Part 2

Identify what's unknown, then solve:

  1. A tank holds 4 times as much water as a 9-liter bucket. Find the tank.
  2. A team's 72 points are 8 times a player's score. Find the player.
  3. A 40-brick wall is how many times as tall as an 8-brick shelf?
Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Same Numbers, Two Different Questions

Problem A: "Mia picked 7 times as many apples as Noah. Noah picked 6 apples. How many did Mia pick?"

Problem B: "Mia picked 7 more apples than Noah. Noah picked 6 apples. How many did Mia pick?"

Solve both before looking at the next slide.

Split diagram: left panel shows 7 copies of a 6-apple bar joined for Mia's total, right panel shows a 6-apple bar extended by 7 more apples

Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Second Pair, Unknown in a New Position

"A bookshelf has 4 times as many books as a table. Bookshelf = 36. Find the table."
36 = 4 × n, n = 9

"A bookshelf has 4 more books than a table. Bookshelf = 36. Find the table."
36 = n + 4, n = 32

Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Strategy: Read the Whole Phrase

  • "Times as many / as much / as long" → multiplicative
  • "More than / fewer than" → additive
  • "More" can hide inside a multiplicative sentence — read the whole phrase

Ask: "How many times, or how many more?"

Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Solo Check: Table and Chair

"A table is 3 times as long as a chair. The table is 27 inches. How long is the chair?"

27 = 3 × n, n = 27 ÷ 3 = 9 inches

Wrong: 3 × 27 = 81 — longer than the table?

Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Capstone, Part 1: Classify, Equation, Solve

Classify, write the equation, then solve:

  1. A cat is 3 times as heavy as a 5 lb rabbit. Find the cat.
  2. A garden has 8 more rows than a 6-row patio. Find the garden.
  3. A jar's 54 marbles are 6 times a cup's amount. Find the cup.
Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Capstone, Part 2: Classify, Equation, Solve

Classify, write the equation, then solve:

  1. A hallway is 9 feet longer than a 4 ft closet. Find it.
  2. A team has 4 times as many wins as its 7 losses.
  3. A shelf holds 32 books; a box holds 4 fewer. Find the box.
Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Four Common Errors to Avoid Today

  1. Always multiplying: check what's unknown first
  2. Mixing up comparison types: "times as many" means multiply, not add
  3. Reversing the reference: the quantity after "as" is the smaller one
  4. Skipping the equation: write it with a symbol before you solve
Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

Key Takeaways From Today's Lesson

✓ Three roles never change: reference, factor, total

✓ One fact family: , ,

✓ Multiplicative or additive comes from the phrase, not one word

✓ Always check reasonableness — does the answer's size fit its role?

Grade 4 Math | 4.OA.A.2
Multiplicative Comparison Word Problems | 4.OA.A.2

What's Next: Multistep Word Problems

4.OA.A.3 — Multistep Problems With All Four Operations

  • A multiplicative comparison becomes one step inside a bigger problem
  • You'll chain two or three operations together to reach a final answer
  • Today's reasonableness check matters even more — more steps, more places for an error to hide
Grade 4 Math | 4.OA.A.2