Problem
Smaller difference means fractions are closer
We have two fractions, 5/8 and 9/10. We want to know which one sits closer to 4/5 on the number line.
Fractions
Your answer
How to solve
Strategy Draw a Diagram — Picturing the three fractions as points on one number line shows that 'closer' just means 'smaller gap'. To compare gaps fairly we break the work into two small subproblems: find the distance from 4/5 to each fraction, then compare the two distances. Rewriting everything over a common denominator lets us measure both gaps with the same ruler.
1STEP 1
Put all three fractions on one ruler
The denominators 8, 10 and 5 share 40, so rewrite all three in fortieths to measure fairly.
4/5=32/40, 5/8=25/40, 9/10=36/40
A 4th-grader knows equal fractions can be renamed; same denominator means same-size pieces, so distances become directly comparable.
4.NF.A.1Draw A DiagramRewriting all three fractions over 40 makes the two distances directly comparable.
Why?
A denominator of 40 cuts the whole into 40 equal pieces, so each fraction becomes a count of pieces that are all the same size.
🧱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?
With one fixed step size, how far apart two fractions sit is simply the difference between their piece counts.
Why?
Every piece lying between the two marks matches exactly one unit of difference between the counts.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
2STEP 2
Measure the gap to 5/8
Find how far 5/8 is from 4/5 by subtracting the two fortieths.
32/40-25/40=7/40
With the same denominator, subtracting is just counting how many 1/40 pieces separate the two points.
5.NF.A.1Identify Subproblems3STEP 3
Measure the gap to 9/10
Find how far 9/10 is from 4/5 the same way.
36/40-32/40=4/40
Again the gap is just the number of equal 1/40 steps between the two points.
5.NF.A.1Identify Subproblems4STEP 4
Compare the two gaps
Now compare the two distances. 4/40 is smaller than 7/40, so 9/10 sits closer to 4/5 than 5/8 does.
4/40 ＜ 7/40
Same denominator means whoever has the smaller numerator is the smaller gap, so that fraction is the closer one.
4.NF.A.2Draw A DiagramAnswer
Both answers are sensible in size: 5/8 = 0.625, 9/10 = 0.9, and 4/5 = 0.8. The decimal gaps are 0.8 - 0.625 = 0.175 and 0.9 - 0.8 = 0.1, matching 7/40 = 0.175 and 4/40 = 0.1. The smaller gap belongs to 9/10, confirming our answer.
Takeaway
Closer just means a smaller gap: measure each distance with the same-size pieces and the tinier gap wins.
- Put all three fractions on one ruler
- Measure the gap to 5/8
- Measure the gap to 9/10
- Compare the two gaps
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested