Questions: Adding Fractions with Like Denominators

Question 1

A student computes 3/8 + 2/8 = 5/16. What mistake did they make?

• A. They added the wrong numerators — the correct numerator sum is 6, giving 6/16
• B. They added both the numerators AND the denominators, but the denominator is a unit label and should not be added
• C. They should have multiplied the fractions, not added them
• D. The problem cannot be solved because the fractions are smaller than one half

Answer: (B)

The student added both numerators (3 + 2 = 5) AND both denominators (8 + 8 = 16), arriving at 5/16. But the denominator is not a quantity to add — it is a label that names the unit (eighths). Adding the denominators is like adding the word 'apples' to itself: '3 apples + 2 apples' does not give you '5 ten-apples.' The pieces didn't change size; you just have more of them. The correct answer is 5/8 — five pieces, each one-eighth of the whole.

Question 2

Why does 2/5 + 1/5 equal 3/5 rather than 3/10?

• A. Because 5 is an odd number, so the denominator stays odd
• B. Because you always keep the larger denominator when adding fractions
• C. Because the denominator names the unit (fifths), and adding more fifths doesn't change the size of each fifth — just like 2 apples + 1 apple = 3 apples, not 3 half-apples
• D. Because 3/10 would be larger than 1 whole

Answer: (C)

The denominator tells you what kind of piece you are working with. When you add 2/5 + 1/5, you have 2 fifths and 1 fifth — three pieces, each one-fifth of the whole. The unit (fifths) does not change just because you added more of them. The answer 3/10 would mean the pieces suddenly became twice as small, which makes no sense — nothing changed the size of each piece. Drawing a fraction bar makes this concrete: two shaded sections plus one shaded section of the same size is clearly three sections of that same size.

Question 3

Adding 3/4 + 3/4 gives an improper fraction that can be converted to the mixed number 1 and 1/2.

• T. True
• F. False

Answer: True

3/4 + 3/4 = 6/4. This is an improper fraction (numerator ≥ denominator). To convert: 4/4 = 1 whole, with 2/4 remaining. 2/4 simplifies to 1/2. So 6/4 = 1 and 2/4 = 1 and 1/2. This is correct. The process — ask how many complete wholes fit, then express the remainder over the same denominator — is the standard conversion method.

Question 4

When adding fractions with the same denominator, you should add both the numerators and the denominators.

• T. True
• F. False

Answer: False

Only the numerators are added; the denominator stays the same. The denominator is a unit label, not a quantity participating in the addition. Adding 2/7 + 3/7 = 5/7, not 5/14. A helpful analogy: '2 inches + 3 inches = 5 inches,' not '5 inches-plus-inches.' The unit label (inches, sevenths) never changes when you add two quantities of the same type.

Question 5

Explain why you add the numerators but keep the denominator when computing 3/7 + 2/7. What does the denominator represent?

Model answer: The denominator (7) names the unit — it tells you each piece is one-seventh of the whole. The numerator counts how many of those pieces you have. When you add 3/7 + 2/7, you are counting: 3 sevenths + 2 sevenths = 5 sevenths = 5/7. The unit (sevenths) does not change; you just have more pieces of the same size.

Understanding denominator-as-unit is the conceptual foundation for all fraction arithmetic. Students who memorize 'add the tops, keep the bottom' without this understanding are vulnerable to errors as soon as fractions appear in new contexts. The unit analogy — sevenths work like inches — makes the rule obvious rather than arbitrary: you never add the unit label, only the count.