Problem
A fraction is a count of unit fractions
Mia paints 1/5 of a poster board every 3 minutes. At that same rate, how long does it take her to paint 3/5 of the board?
Fractions
Your answer
minutes
Watch out Don't compute the time for the whole board (5 × 3 = 15) - the question asks for 3/5, not all 5/5.
How to solve
Strategy Look for a Pattern — Seeing 3/5 as three units of 1/5 turns the problem into repeated equal chunks: each 1/5 costs 3 minutes, so the total is just that time pattern repeated three times. A bar drawing makes the equal chunks obvious.
1STEP 1
See 3/5 as three 1/5s
The fraction 3/5 is three copies of the unit fraction 1/5. So painting 3/5 means painting 1/5 three separate times.
3/5 = 1/5 + 1/5 + 1/5
A fraction is just a count of unit fractions, so 3/5 is 'three fifths' literally.
4.NF.B.3Draw A DiagramThe fraction 3/5 is three copies of the unit fraction 1/5, that is, 3/5 = 1/5 + 1/5 + 1/5.
Why?
When one whole board is cut into 5 equal parts, three of those parts placed together with no gaps or overlaps make exactly the 3/5 region, so 3/5 is those three parts added up.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Each of the three parts is the same size, one 1/5, so having three of them is the same amount as one 1/5 counted three times over.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Each fifth costs 3 minutes
Painting one 1/5 takes 3 minutes, and there are three of them, so multiply 3 minutes by 3.
3 × 3 = 9
Equal-size pieces each take the same time, so three pieces take three times as long.
3.NF.A.1Look For A PatternAnswer
9 minutes
3 × 3 = 9
3/5 is three times as much board as 1/5, so it should take three times as long as 3 minutes, which is 9 minutes. That is less than the time for the whole board (5 fifths would be 15 minutes), so 9 minutes is reasonable.
Takeaway
A fraction is just a count of unit fractions, so 3/5 takes three times the time of one 1/5!
- See 3/5 as three 1/5s
- Each fifth costs 3 minutes
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested