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Fractions as Numbers: Equivalent Fractions, Compare, Number Line
Build Grade 4–5 fraction sense by locating fractions on number lines, creating equivalents, comparing values, and moving between improper fractions and mixed numbers.
The idea
A fraction is a number, not just a picture of shaded pieces. In a/b, the denominator b tells how many equal parts make one whole, and the numerator a tells how many of those parts you have. Fractions belong on the same number line as whole numbers.
Equivalent fractions name the same point. Multiplying or dividing the numerator and denominator by the same non-zero number changes the names of the pieces and how many pieces are counted, but not the amount. For example, 3/4 = 6/8 = 9/12.
To compare fractions, use structure before reaching for a rule. Same denominators: compare numerators. Same numerators: fewer equal pieces means larger pieces. Otherwise, use a benchmark such as 1/2 or 1, make a common denominator, or cross-check with multiplication. Fractions greater than 1 can also be written as mixed numbers.
Worked examples
Example 1 · Put a fraction on a number line
Where is 5/8 between 0 and 1?
- Divide the distance from 0 to 1 into 8 equal intervals.
- Count 5 intervals from 0.
- That point is 5/8, which is just to the right of 1/2 because 1/2 = 4/8.
5/8 is the fifth eighth-mark from 0, one eighth past 1/2.
Example 2 · Build an equivalent fraction
Write 3/5 with denominator 20.
- 5 × 4 = 20, so multiply the denominator by 4.
- Multiply the numerator by the same 4: 3 × 4 = 12.
- The value stays the same because both parts were scaled by the same factor.
3/5 = 12/20.
Example 3 · Simplify without changing value
Simplify 18/24.
- 18 and 24 are both divisible by 6.
- 18 ÷ 6 = 3 and 24 ÷ 6 = 4.
- Dividing numerator and denominator by the same factor preserves the value.
18/24 = 3/4.
Example 4 · Compare with a common denominator
Which is larger: 7/12 or 5/8?
- A common denominator is 24.
- 7/12 = 14/24 and 5/8 = 15/24.
- 15/24 is larger than 14/24.
5/8 is larger by 1/24.
Example 5 · Use a benchmark
Compare 7/12 with 1/2.
- Write 1/2 as twelfths: 1/2 = 6/12.
- 7/12 is one twelfth more than 6/12.
- So 7/12 is greater than 1/2.
7/12 > 1/2.
Example 6 · Cross 1 whole
Write 11/4 as a mixed number.
- Four fourths make 1 whole.
- 11 fourths contain 8 fourths, or 2 wholes, with 3 fourths left.
- So 11/4 and 2 3/4 name the same point on the number line.
11/4 = 2 3/4.
Common mistakes
These are tempting because they use familiar operations. Check what the question is actually measuring.
Why this fails: Equivalence comes from multiplying or dividing both by the same non-zero factor. For example, 8/12 = 2/3, but subtracting 4 gives 4/8 = 1/2, which is a different number.
Why this fails: For the same whole, a larger denominator means the whole was cut into more equal pieces, so each piece is smaller.
Why this fails: The numerator counts pieces, but the denominator tells the size of those pieces. You need a common basis for comparison.
Why this fails: Fractions can be greater than 1. A value such as 9/4 is simply 2 1/4 on the number line.
Why this fails: Cross-products can compare two positive fractions, but benchmark and common-denominator reasoning make the size relationship easier to explain and check.
Foundation
Foundation practice
Read fraction notation, make simple equivalents, use benchmarks, and connect fractions to the number line.
1. In 5/8, what does the denominator 8 tell you?
2. 3/5 = ?/20. Find the missing numerator.
3. Simplify 4/6.
4. From 0 to 1, a segment is divided into 8 equal intervals. What fraction is the point 6 intervals from 0? Give a simplified form too.
5. Are 2/3 and 8/12 equivalent?
- Yes
- No
6. Which is larger: 3/8 or 5/8?
7. Which is larger: 4/7 or 4/9?
8. Is 7/12 less than, equal to, or greater than 1/2?
9. Write 9/4 as a mixed number.
10. Write 1 3/5 as an improper fraction.
Core
Core practice
Compare unlike fractions, solve missing-value equivalence problems, and explain fraction size in context.
11. Which is larger: 5/6 or 7/9?
12. Which is larger: 7/12 or 5/8?
13. Order 1/2, 3/4, and 2/3 from least to greatest.
14. 9/12 = ?/4. Find the missing numerator.
15. 5/7 = 20/?. Find the missing denominator.
16. Mina walks 3/4 km and Theo walks 5/8 km. Who walks farther, and by how much?
17. Simplify 14/20.
18. Which fraction lies between 1/2 and 3/4?
- 3/8
- 5/8
- 7/8
- 4/3
19. Write 11/6 as a mixed number.
20. Write 2 2/3 as an improper fraction.
Think
Think practice
Spot false comparison shortcuts, work backwards from equivalent fractions, and justify close comparisons.
21. Are 6/8 and 9/12 equivalent? Explain briefly.
22. A student says 3/5 < 3/7 because 5 < 7. Is the student correct?
23. A fraction equivalent to 3/4 has denominator 28. What is the numerator?
24. A fraction equivalent to 4/7 has numerator 16. What is its denominator?
25. n/18 = 5/6. Find n.
26. Which is larger: 11/12 or 8/9?
27. Order 7/10, 2/3, and 3/4 from least to greatest.
28. One route is 6/8 km and another is 9/12 km. Are the routes the same length?
29. Why does multiplying both numerator and denominator by 5 keep a fraction's value unchanged?
30. Which is closest to 1: 7/8, 9/10, 11/12, or 13/15?
Challenge
Challenge practice
Use fraction structure for between-number problems, close comparisons, reverse equivalence, and an early algebra preview.
31. Find a fraction with denominator 24 that lies strictly between 2/3 and 3/4.
32. Are 18/24 and 21/28 equivalent?
33. A fraction equals 4/5 and its denominator is 35. What is its numerator?
34. 2 1/3 = q/6. Find q.
35. Which is larger: 5/6 or 14/17?
36. Which is larger: 13/20 or 2/3?
37. A fraction equivalent to 7/9 has a numerator greater than 30. What is the smallest possible numerator, and what is the fraction?
38. A positive fraction has numerator 23 and is less than 1. What is the smallest whole-number denominator it can have?
39. A student 'simplifies' 8/12 by subtracting 4 from both parts and gets 4/8. Explain the error and give the correct simplified fraction.
40. If n/24 = 5/8, find n.
Answer key
- 1. One whole is divided into 8 equal parts.
The denominator names the size of the equal parts; the numerator says five of those eighths are counted. - 2. 12
5 × 4 = 20, so multiply 3 by the same factor: 3 × 4 = 12. - 3. 2/3
Divide numerator and denominator by their common factor 2. - 4. 6/8 = 3/4
The point counts 6 eighths. Dividing 6 and 8 by 2 gives 3/4. - 5. Yes
2/3 × 4/4 = 8/12, so they name the same value. - 6. 5/8
The denominators match, so compare numerators: 5 eighths is more than 3 eighths. - 7. 4/7
With the same numerator, sevenths are larger pieces than ninths, so 4/7 > 4/9. - 8. Greater than 1/2
1/2 = 6/12, and 7/12 is one twelfth larger. - 9. 2 1/4
8/4 makes 2 wholes, leaving 1/4. - 10. 8/5
One whole is 5/5; 5/5 + 3/5 = 8/5. - 11. 5/6
Using denominator 18, 5/6 = 15/18 and 7/9 = 14/18. - 12. 5/8
Using denominator 24, 7/12 = 14/24 and 5/8 = 15/24. - 13. 1/2 < 2/3 < 3/4
With denominator 12, the fractions are 6/12, 8/12, and 9/12. - 14. 3
9/12 simplifies by dividing both parts by 3, giving 3/4. - 15. 28
The numerator was multiplied by 4, so the denominator must also be multiplied by 4: 7 × 4 = 28. - 16. Mina, by 1/8 km
3/4 = 6/8, which is 1/8 more than 5/8. - 17. 7/10
Divide 14 and 20 by their common factor 2. - 18. 5/8
1/2 = 4/8 and 3/4 = 6/8, so 5/8 lies between them. - 19. 1 5/6
6/6 makes one whole, leaving 5/6. - 20. 8/3
Two wholes are 6/3; add 2/3 to get 8/3. - 21. Yes; both equal 3/4.
6/8 simplifies to 3/4, and 9/12 also simplifies to 3/4. - 22. No; 3/5 > 3/7.
With the same numerator, fifths are larger pieces than sevenths, so three fifths is larger. - 23. 21
4 × 7 = 28, so 3 × 7 = 21. - 24. 28
4 × 4 = 16, so 7 × 4 = 28. - 25. 15
6 × 3 = 18, so 5 × 3 = 15. - 26. 11/12
Using denominator 36, 11/12 = 33/36 and 8/9 = 32/36. - 27. 2/3 < 7/10 < 3/4
With denominator 60, the values are 40/60, 42/60, and 45/60. - 28. Yes; both are 3/4 km.
Each fraction simplifies to 3/4. - 29. It renames each original part as 5 smaller equal parts and counts 5 times as many of them, so the total amount stays the same.
Scaling both parts by the same factor multiplies the fraction by 5/5, which equals 1. - 30. 11/12
Their gaps from 1 are 1/8, 1/10, 1/12, and 2/15. The smallest gap is 1/12. - 31. 17/24
2/3 = 16/24 and 3/4 = 18/24, so 17/24 lies between them. - 32. Yes; both equal 3/4.
18/24 divides by 6 to 3/4, and 21/28 divides by 7 to 3/4. - 33. 28
5 × 7 = 35, so 4 × 7 = 28. - 34. 14
2 1/3 = 7/3, and multiplying numerator and denominator by 2 gives 14/6. - 35. 5/6
Cross-products are 5 × 17 = 85 and 14 × 6 = 84, so 5/6 is slightly larger. - 36. 2/3
Cross-products are 13 × 3 = 39 and 2 × 20 = 40, so 2/3 is larger. - 37. 35; the fraction is 35/45.
Equivalent numerators are multiples of 7. The first multiple above 30 is 35 = 7 × 5, so the denominator is 9 × 5 = 45. - 38. 24
For a positive fraction to be less than 1, the denominator must be greater than the numerator. The smallest whole number greater than 23 is 24. - 39. Subtracting the same number does not preserve a fraction's value; 8/12 simplifies to 2/3.
Divide numerator and denominator by the common factor 4: 8 ÷ 4 over 12 ÷ 4 = 2/3. The student's 4/8 equals 1/2. - 40. 15
24 is 3 times 8, so n must be 3 times 5: n = 15.