Add or Subtract Mixed Numbers in a Word Problem - ISEE Lower Level Mathematics Achievement

SECTION 1

Why Do We Need Mixed Numbers?

People have been using fractions for thousands of years! Ancient Egyptians used fractions to measure land and share food. But sometimes a fraction alone was not enough.

Imagine you ate one whole pizza and half of another. You would not say just "one half." You would say "one and a half." That is a mixed number — a whole number and a fraction together.

3000 BC • Ancient Egypt

Egyptians used fractions to divide bread and measure fields along the Nile River.

500 AD • India

Indian mathematicians wrote fractions the way we do today, with a numerator on top and a denominator on the bottom.

1200 AD • Europe

Europeans started using mixed numbers in trade and cooking, combining whole amounts with fractional parts.

Today • Your ISEE Test

Mixed numbers appear in word problems about recipes, distances, time, and more. You will see them on the ISEE!

The big question is: how do you add or subtract mixed numbers when they show up in a story problem? Let's find out!

SECTION 2

Core Ideas You Need to Know

Before we solve word problems, let's review a few important ideas. These are the building blocks you will use every time.

What Is a Mixed Number?

A mixed number has a whole-number part and a fraction part. For example, 3½ means 3 wholes and ½ more.

Like Denominators

When fractions have the same bottom number (denominator), they are called like denominators. You can add or subtract them right away!

Add or Subtract in Parts

Add the whole numbers together. Then add the fractions together. It is like stacking blocks — big pieces first, small pieces next.

Borrowing (Regrouping)

Sometimes the fraction you are subtracting is bigger. You "borrow" 1 from the whole number and turn it into a fraction. This is called regrouping.

Read the Story Carefully

Word problems use clue words. "In all" or "total" means add. "How much more" or "left" means subtract. Circle these words!

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KEY TAKEAWAY

Think of mixed numbers like coins. The whole number is like dollar bills, and the fraction is like loose change. When you add, you combine the bills first, then the change. When you subtract, you take away bills, then change. If you do not have enough change, you break a dollar into coins (that is borrowing!).

SECTION 3

See It: Adding Mixed Numbers

Pictures make mixed numbers much easier! Look at the diagram below. It shows how to add 2¾ + 1¼ using fraction bars.

The purple bars show 2¾. The blue bars show 1¼. The ¾ and ¼ combine to make another whole. The green bars at the bottom show the answer: 4.

See how ¾ and ¼ fit together to make one more whole? That is why 2¾ + 1¼ = 4, not 3 and something. The fraction pieces filled up to make a complete bar!

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ISEE Tip

On the ISEE, draw a quick picture in your test booklet. Even a small sketch of bars or circles can help you see the answer clearly. You are allowed to write in the booklet!

SECTION 4

The Step-by-Step Method

Here are the simple steps you can follow every single time. Think of it as a recipe!

ADDING MIXED NUMBERS

a b/d + c e/d = (a + c) (b + e)/d

Add the whole numbers (a + c). Then add the fractions (b/d + e/d). If the fraction part is improper, regroup it.

SUBTRACTING MIXED NUMBERS

a b/d − c e/d = (a − c) (b − e)/d

Subtract the whole numbers. Then subtract the fractions. If the top fraction is too small, borrow 1 from the whole number first.

Four Easy Steps

Step 1 — Read the problem. Find the mixed numbers and decide if you add or subtract.
Step 2 — Work with the whole numbers. Add them or subtract them.
Step 3 — Work with the fractions. Add or subtract the numerators. Keep the denominator the same.
Step 4 — Simplify if needed. If the fraction is improper (top ≥ bottom), turn the extra into a whole number.

ℹ

What About Borrowing?

If you need to subtract and the first fraction is smaller, borrow 1 from the whole number. Turn that 1 into a fraction with the same denominator. For example, if you have 5⅓ − 2⅔, change 5⅓ into 4 and 4/3. Now ⁴⁄₃ − ⅔ = ²⁄₃. The answer is 2⅔.

SECTION 5

Clue Words in Word Problems

Word problems can be tricky. The secret is to spot the clue words that tell you what to do. Let's sort them out!

Green box: clue words that tell you to add. Pink box: clue words that tell you to subtract. Circle the clue words when you read a problem!

Before you do any math, underline the numbers and circle the clue words. This one simple habit will help you avoid silly mistakes on the ISEE!

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ISEE Strategy

On the ISEE there is no penalty for wrong answers. Always pick an answer, even if you are not sure. Use the clue words and try to eliminate at least two choices. Then guess from what is left!

SECTION 6

Worked Example: Step by Step

Let's solve a full word problem together. Follow along with each step!

Problem: Maria walked 3⅖ miles on Saturday and 2⅗ miles on Sunday. How many miles did she walk in all?

Solving: 3⅖ + 2⅗

1
Step 1 — Read and Find Clue Words

The problem says "in all". That means we add. The two mixed numbers are 3⅖ and 2⅗.
2
Step 2 — Add the Whole Numbers

The whole-number parts are 3 and 2.

3 + 2 = 5
3
Step 3 — Add the Fractions

The fractions are ⅖ and ⅗. They have the same denominator (5), so just add the numerators: 2 + 3 = 5.

⅖ + ⅗ = ⁵⁄₅
4
Step 4 — Simplify

⁵⁄₅ equals 1 whole. Add that extra 1 to the 5 we already have: 5 + 1 = 6.

Maria walked 6 miles in all.

Great job! Notice how we handled the fractions separately from the whole numbers. When the fraction added up to a whole, we regrouped it. You've got this!

SECTION 7

Common Mistakes to Avoid

Even strong students make these mistakes. Let's learn about them now so they do not catch you on test day!

Mistake	Why It Happens	How to Fix It
Adding denominators	Students add ⅖ + ⅗ and write ⁵⁄₁₀ instead of ⁵⁄₅.	Never add the bottom numbers! Keep the denominator the same.
Forgetting to regroup	They get 5⁵⁄₅ and leave it like that.	If the numerator equals or is bigger than the denominator, turn it into a whole number.
Forgetting to borrow	In 5⅓ − 2⅔, they try to do ⅓ − ⅔ and get confused.	Borrow 1 from the 5. Change 5⅓ to 4⁴⁄₃. Now subtract.
Choosing add vs. subtract wrong	They do not read the clue words carefully.	Circle clue words first. Ask: am I combining or taking away?

Top mistakes and how to beat them

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REMEMBER

The denominator is like the size of your pizza slices. If you and your friend both cut your pizzas into 5 slices, you count the total slices but the slice size stays the same. You never add the denominators!

SECTION 8

What Comes Next?

Right now on the ISEE Lower Level, your mixed-number problems will use like denominators (the same bottom number). But as you grow in math, you will also work with unlike denominators. Here is a quick peek.

What You Know Now	What You Will Learn Later
Fractions with the same denominator (like ⅖ + ⅗)	Fractions with different denominators (like ½ + ⅓)
Add or subtract the numerators directly	Find a common denominator first, then add or subtract
Simple regrouping	Multiplying and dividing mixed numbers

Your skills will keep growing!

For now, master the basics. The skills you build here are the foundation for everything ahead. You are doing amazing work!

SECTION 9

Practice Problems

Time to practice! Read each problem carefully, circle the clue words, and pick the best answer. Remember, on the real ISEE, always answer every question — no points are taken off for wrong answers.

Question 1: CONCEPTUAL

Sam ate 1³⁄₈ slices of pie and then ate 2²⁄₈ more slices. How many slices of pie did Sam eat in all?

A. 3¼
B. 3⅜
C. 3⅝
D. 4⅛

Show answer (C)

The clue words "in all" tell us to add. Whole numbers: 1 + 2 = 3. Fractions: ³⁄₈ + ²⁄₈ = ⁵⁄₈. So the answer is 3⅝. Choice A (3¼) is wrong because ¼ = ²⁄₈, which is just the second fraction alone. Choice B (3⅜) is wrong because ⅜ is only the first fraction. Choice D (4⅛) is wrong because it adds the denominators (8 + 8 = 16, then simplifies incorrectly).

Question 2: BASIC CALCULATION

A ribbon is 6⁴⁄₆ feet long. Maya cuts off 2¹⁄₆ feet. How many feet of ribbon are left?

A. 4¹⁄₆
B. 4⅓
C. 4½
D. 5½

Show answer (C)

"Left" is a subtraction clue word. Whole numbers: 6 − 2 = 4. Fractions: ⁴⁄₆ − ¹⁄₆ = ³⁄₆. Simplify ³⁄₆ to ½. The answer is 4½. Choice A (4¹⁄₆) uses only the second fraction instead of finding the difference. Choice B (4⅓) subtracts too many sixths — ³⁄₆ simplified is ½, not ⅓. Choice D (5½) adds the whole numbers instead of subtracting them.

Question 3: INTERMEDIATE

Jake practiced piano for 1³⁄₅ hours on Monday and 2⁴⁄₅ hours on Tuesday. How many hours did he practice altogether?

A. 3⅖
B. 3⅘
C. 4⅕
D. 4⅖

Show answer (D)

"Altogether" means add. Whole numbers: 1 + 2 = 3. Fractions: ³⁄₅ + ⁴⁄₅ = ⁷⁄₅. Since ⁷⁄₅ is improper, convert: ⁷⁄₅ = 1²⁄₅. Add the extra 1 to 3: 3 + 1²⁄₅ = 4²⁄₅. Choice A (3⅖) forgets to regroup the improper fraction. Choice B (3⅘) just uses the bigger fraction. Choice C (4⅕) regroups but writes the wrong numerator.

Question 4: APPLIED

A recipe needs 4²⁄₄ cups of flour. Tina already put in 1³⁄₄ cups. How many more cups does she need?

A. 2¼
B. 2½
C. 2¾
D. 3¼

Show answer (C)

"How many more" means subtract: 4²⁄₄ − 1³⁄₄. The fraction ²⁄₄ is smaller than ³⁄₄, so we borrow. Change 4²⁄₄ to 3⁶⁄₄ (borrow 1 whole = ⁴⁄₄, add to ²⁄₄ to get ⁶⁄₄). Now: 3 − 1 = 2 and ⁶⁄₄ − ³⁄₄ = ³⁄₄. The answer is 2¾. Choice A (2¼) subtracts the fractions backwards without borrowing: ³⁄₄ − ²⁄₄ = ¼. Choice B (2½) makes an error during simplification. Choice D (3¼) does not borrow correctly from the whole number.

Question 5: CRITICAL THINKING

Ella walked 6⅖ miles on a hiking trail. Her friend Mia walked 5⅕ miles on the same trail. How much farther did Ella walk than Mia?

A. ⅕
B. 1
C. 1⅕
D. 1⅖

Show answer (C)

Subtract the mixed numbers: 6⅖ − 5⅕. The fractions have the same denominator, so subtract the fractions first: ⅖ − ⅕ = ⅕. Then subtract the whole numbers: 6 − 5 = 1. The answer is 1⅕. Choice A (⅕) only subtracts the fraction parts and forgets the whole numbers. Choice B (1) only subtracts the whole numbers and forgets the fraction parts. Choice D (1⅖) adds the fractions instead of subtracting them. Remember to subtract both the whole number part and the fraction part!

SUMMARY

Lesson Summary

Summary

A mixed number has a whole-number part and a fraction part. To add mixed numbers, add the whole numbers, then add the fractions. If the fraction becomes improper, regroup it into an extra whole. To subtract mixed numbers, subtract the whole numbers and fractions. If the top fraction is too small, borrow 1 from the whole number first.

Always look for clue words in word problems: "in all" and "total" mean add; "how much more" and "left" mean subtract. Never add the denominators — keep them the same! On the ISEE, always answer every question and use process of elimination if you are unsure. You've got this!