Concept 5 of 9
Two Ways to Say It: Mixed Numbers and Improper Fractions
Today's big idea
A mixed number names the wholes; an improper fraction counts every slice.
1 3/4 and 7/4 are one pizza and three quarters, counted two ways.
How it works
👀 What's going on?
You cut two pizzas into quarters and eat one slice. What is left can be said two ways. One whole pizza and three quarters, or simply seven quarter-slices. Nothing has moved. Only the way of counting has changed.
💡 Here's the trick
Look at the plates above. A mixed number names the wholes and then the leftover part, like 1 3/4. An improper fraction ignores the wholes and counts every single slice, like 7/4, so its top is bigger than its bottom. To go from mixed to improper, multiply the whole by the bottom and add the top. That is 1 x 4 = 4 slices in the whole pizza, plus the 3 already loose, giving 7/4. To come back, divide: 7 divided by 4 is 1 with 3 left over, which is 1 3/4. The bottom number never changes, because the slices stay the same size all the way through.
🪄 How to do it
- 1.Going to improper, multiply the whole number by the bottom of the fraction, as the plates above do.
- 2.Add the top number on, and keep the same bottom.
- 3.Coming back, divide the top by the bottom: the answer is the whole and the leftover is the new top.
✍️ Try it
Turn 2 3/5 into an improper fraction. Then turn 11/4 back into a mixed number.
For the first, 2 x 5 = 10 fifths in the wholes, plus the 3 already there, giving 13/5. For the second, 11 divided by 4 is 2 with 3 over, so it is 2 3/4. Both kept the bottom number they started with.
⚠️ Watch out
You might be thinking 2 3/5 means 2 multiplied by 3/5. It does not. The whole and the fraction are added, so 2 3/5 is 2 + 3/5. Reading it as a multiplication gives 6/5, which is barely more than one, when the real answer is over two and a half.
🌟 Why this matters
A recipe saying 7/4 cups and one saying 1 3/4 cups want the same jug filled the same way. Being able to swap is also what lets you take 1 2/5 away from 3 1/5, because the wholes can be broken into fifths first.
Exam tip: Convert both numbers to improper form before comparing them, since 9/4 against 2 1/8 is far easier once both are quarters and eighths.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“Write 2 3/5 as an improper fraction.”
- 1
Multiply whole by the bottom
2 wholes × 5 fifths = 10 fifths.
- 2
Add the top
10 + 3 = 13
- 3
Keep the same bottom
All the pieces are still fifths, so the bottom stays 5.
- 4
Answer ✓
2 3/5 = 13/5
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
ON THE PLATES
1 3/4
1 whole pizza and 3 quarter-slices.
Quick check
✅ VERIFICATION
Which improper fraction is the same as 3 2/5?
⚡ Finish the activity above first to answer.
Keep going!
Up next: The Sorting Trays
Split wholes and fractions, then glue back.