Identify a fraction between two given fractions. - ISEE Lower Level Mathematics Achievement

SECTION 1

Where Did Fractions Come From?

Have you ever split a pizza with a friend? That's exactly the kind of problem that led people to invent fractions! Thousands of years ago, people needed a way to talk about parts of a whole thing.

Over time, people realized they also needed to figure out what goes between two fractions. Imagine you know one piece of rope is 1/4 of a yard and another is 3/4 of a yard. What length is in the middle? Let's see how this idea grew!

3000 BC • Ancient Egypt

Egyptians used fractions to measure land and share bread fairly. They mostly used fractions with a 1 on top, like 1/2, 1/3, and 1/4.

500 BC • Ancient Greece

Greek thinkers discovered that between any two fractions, you can always find another fraction. There are endless fractions hiding in between!

600 AD • India

Mathematicians in India invented the fraction bar we use today. They wrote fractions with a top number over a bottom number, just like we do.

Today • ISEE & School Math

Now you use fractions every day in math class. The ISEE test asks you to find fractions that fit between two other fractions!

So here's the big question: if someone gives you two fractions, how do you find one that goes right in between them? That's exactly what this lesson will teach you. Let's go!

SECTION 2

Key Ideas About Fractions in Between

Before we start finding fractions in between, let's review some important ideas. These are your tools for solving these problems!

Common Denominators

When fractions have the same bottom number (denominator), you can compare them easily. Just look at the top numbers! For example, 2/5 is less than 4/5.

Equivalent Fractions

You can rename a fraction by multiplying top and bottom by the same number. 1/2 = 2/4 = 3/6. They are all the same size!

Fractions on a Number Line

Every fraction has a spot on a number line. A fraction between two others sits in the space between their spots.

Bigger Denominators Make More Spaces

When you use a bigger denominator, you create more spaces between 0 and 1. More spaces means more fractions to choose from!

✦

KEY TAKEAWAY

Think of fractions like seats on a bus. If the bus has 4 seats (denominator = 4), you can only sit in 4 spots. But if you switch to a bus with 8 seats (denominator = 8), now you have more spots to choose from. That's how you find a fraction between two others — you make more spaces!

SECTION 3

See It on a Number Line

A number line is your best friend for this topic! Let's look at fractions between 1/4 and 3/4. The diagram below shows how we can find fractions that sit between them.

The top number line shows fourths. The bottom number line shows eighths. By using a bigger denominator (8 instead of 4), we created more spaces and found 3/8, 4/8, and 5/8 between 1/4 and 3/4.

See how the second number line has more tick marks? That's the magic trick. When you use a bigger denominator, you create more spots on the number line. More spots means more fractions to pick from!

SECTION 4

How to Find a Fraction in Between

Here are two strategies you can use on the ISEE. Pick the one that feels easiest to you!

Strategy 1: Use Common Denominators

STEP-BY-STEP

Rewrite both fractions with the same denominator → Look for a top number in between

Example: Find a fraction between 1/3 and 2/3. They already have the same denominator (3), but there's no whole number between 1 and 2. So multiply top and bottom of each by 2: 1/3 = 2/6 and 2/3 = 4/6. Now 3/6 is right in between!

Strategy 2: Double the Denominator

QUICK TRICK

Multiply the top and bottom of BOTH fractions by 2

This doubles the denominator and creates new spaces. Example: Between 2/5 and 3/5 → Rewrite as 4/10 and 6/10. Now you can pick 5/10 (which is the same as 1/2). Easy!

💡

ISEE Test Tip!

On the ISEE, you usually don't have to calculate the answer from scratch. Instead, you check which answer choice fits between the two given fractions. Try each answer choice! Convert everything to the same denominator and see which one lands in the middle.

Remember: on the ISEE, there is no penalty for guessing. Always pick an answer, even if you're not sure. Cross out choices that are too small or too big first!

SECTION 5

Comparing Fractions to Check Your Answer

After you pick a fraction, you need to make sure it really is between the two given fractions. Here's how to check. The diagram below shows the whole process!

This flowchart shows the three steps: find a common denominator, convert all fractions, then compare the numerators (top numbers). If the middle fraction's top number is between the other two, you found your answer!

Find a common denominator — pick a number that all the denominators divide into evenly.
Rewrite each fraction — multiply top and bottom by the same number.
Compare the top numbers — if the middle fraction's numerator is between the other two, it works!

SECTION 6

Let's Work Through One Together!

Here's a problem just like the ones on the ISEE. Let's solve it step by step. You've got this!

📝

Problem

Which fraction is between 1/3 and 5/6? (A) 1/4 (B) 1/2 (C) 5/12 (D) 7/8

Step-by-Step Solution

1
Step 1 — Find a Common Denominator

The fractions in the problem use denominators 3, 6, 4, 2, 12, and 8. A number that all of these divide into evenly is 24. But let's try 12 first since most of these fit. 3 goes into 12, 6 goes into 12, 4 goes into 12, 2 goes into 12, and 12 goes into 12. Only 8 doesn't go evenly into 12, so we might need to use 24 for choice D.

Common denominator: 12 (and 24 for checking D)
2
Step 2 — Convert the Given Fractions

1/3 = 4/12 (multiply top and bottom by 4). 5/6 = 10/12 (multiply top and bottom by 2). So we need a fraction with a top number between 4 and 10 when the denominator is 12.

We need a numerator between 4 and 10 out of 12.
3
Step 3 — Check Each Answer Choice

(A) 1/4 = 3/12. Is 3 between 4 and 10? No, 3 is too small. ✗ (B) 1/2 = 6/12. Is 6 between 4 and 10? Yes! ✓ (C) 5/12 = 5/12. Is 5 between 4 and 10? Yes! ✓ (D) 7/8 = 21/24. Convert our fractions too: 1/3 = 8/24 and 5/6 = 20/24. Is 21 between 8 and 20? No, 21 is too big. ✗

Both B and C work, but we pick one answer.
4
Step 4 — Pick the Best Answer

On a real ISEE test, only one answer choice will be correct. In this example, both B (1/2) and C (5/12) are between 1/3 and 5/6. On the real test, only one will appear. The key is knowing how to check each choice using common denominators.

Both B and C are between 1/3 and 5/6!

SECTION 7

Strategies and Common Mistakes

Let's look at what works well and what trips kids up on this type of question. Knowing these will help you avoid mistakes on test day!

Strategy	Why It Helps	Watch Out For
Common Denominators	Makes comparing easy — just look at top numbers	Don't forget to multiply BOTH the top and bottom by the same number
Use Benchmarks	Compare to 0, 1/2, and 1 to estimate quickly	Estimating alone might not be exact enough — always check
Eliminate Wrong Choices	Crossing out answers that are too big or too small saves time	Don't assume a fraction is bigger just because its numbers look bigger (3/8 is less than 1/2)
Draw a Number Line	You can see where each fraction sits	Takes time — use it only when you're stuck

Strategies for Finding Fractions in Between

⚠️

AVOID THIS COMMON TRAP

A bigger denominator does NOT mean a bigger fraction! Think of it like pizza slices. If you cut a pizza into 8 slices, each slice is smaller than if you cut it into 4 slices. So 1/8 is less than 1/4, even though 8 is bigger than 4.

SECTION 8

From Fractions in Between to Bigger Ideas

Finding a fraction between two fractions is a skill you'll use again and again. Let's see how this idea connects to things you'll learn later!

What You Know Now	What Comes Next
Find a fraction between two fractions	Find the exact middle (average) of two fractions
Use common denominators to compare	Add and subtract fractions with unlike denominators
Understand equivalent fractions	Simplify fractions to lowest terms
Place fractions on a number line	Place decimals and mixed numbers on a number line

How This Skill Connects to Future Math

Here's a cool fact: between any two fractions, there are infinitely many other fractions. You can always keep doubling the denominator to find more! But for the ISEE, you just need to find one. 😊

SECTION 9

Practice Problems

Time to practice! Try each problem, then check your answer. Remember to use common denominators and elimination. You've got this!

Question 1: CONCEPTUAL

Which fraction is between 1/4 and 3/4?

A. 1/8
B. 1/2
C. 7/8
D. 1

Show answer (B)

The correct answer is B (1/2). Convert to eighths: 1/4 = 2/8, 1/2 = 4/8, 3/4 = 6/8. Since 2 < 4 < 6, 1/2 is between 1/4 and 3/4. Choice A (1/8) is too small — it's less than 1/4. Choice C (7/8) is too big — it's more than 3/4. Choice D (1) is a whole number and is bigger than 3/4.

Question 2: BASIC CALCULATION

Which fraction is between 1/6 and 5/6?

A. 1/12
B. 1/3
C. 11/12
D. 1

Show answer (B)

The correct answer is B (1/3). Convert to sixths: 1/3 = 2/6. Now check: is 2/6 between 1/6 and 5/6? Yes, because 1 < 2 < 5. Choice A (1/12) equals about half of 1/6, so it's too small. Choice C (11/12) is almost 1, so it's bigger than 5/6. Choice D (1) is bigger than 5/6.

Question 3: INTERMEDIATE

Which fraction is between 1/3 and 1/2?

A. 1/6
B. 1/4
C. 5/12
D. 7/12

Show answer (C)

The correct answer is C (5/12). Convert all fractions to twelfths: 1/3 = 4/12 and 1/2 = 6/12. A fraction between them must have a numerator between 4 and 6 when written in twelfths. That gives us 5/12, since 4/12 < 5/12 < 6/12. Choice A (1/6 = 2/12) is less than 1/3 because 2 < 4, so it is too small. Choice B (1/4 = 3/12) is less than 1/3 because 3 < 4, so it is also too small. Choice D (7/12) is greater than 1/2 because 7 > 6, so it is too large.

Question 4: APPLIED

Maya ran 2/5 of a mile. Josh ran 4/5 of a mile. Which fraction of a mile is between the distances they ran?

A. 1/5
B. 3/10
C. 3/5
D. 9/10

Show answer (C)

The correct answer is C (3/5). We need a fraction between 2/5 and 4/5. Since they already have the same denominator, just look at the top numbers: we need something between 2 and 4. That's 3, so 3/5 works! Choice A (1/5) has a numerator of 1, which is less than 2, so it's too small. Choice B (3/10 = 3/10) — convert 2/5 = 4/10 and 3/10 < 4/10, so it's too small. Choice D (9/10 = 9/10) — convert 4/5 = 8/10 and 9 > 8, so it's too big.

Question 5: CRITICAL THINKING

Which fraction is between 3/8 and 5/8?

A. 1/4
B. 7/16
C. 3/4
D. 13/16

Show answer (B)

The correct answer is B (7/16). Convert 3/8 and 5/8 to sixteenths: 3/8 = 6/16 and 5/8 = 10/16. We need a fraction with a numerator between 6 and 10. Choice B is 7/16, and 6 < 7 < 10, so 7/16 is between 6/16 and 10/16. ✓ Choice A (1/4 = 4/16) is less than 6/16, so it is too small. Choice C (3/4 = 12/16) is greater than 10/16, so it is too big. Choice D (13/16) is also greater than 10/16, so it is too big.

SUMMARY

Let's Wrap It All Up!

Summary

To find a fraction between two given fractions, start by writing both fractions with a common denominator. If the numerators are right next to each other (like 2 and 3), double the denominator to create more spaces. Then pick a fraction whose top number falls between the other two top numbers.

On the ISEE, use process of elimination to test each answer choice. Convert everything to the same denominator and compare. Cross out choices that are too small or too big. Remember, there is no penalty for guessing, so always answer every question. You've got this!