Problem
Compare fractions sharing numerator or denominator
I need to count how many of the digits 1 through 9 can be placed in the star so that the fraction 1/7 is less than 1/star.
Fractions
Your answer
Watch out It's tempting to think star must be greater than 7 since 7 is the bigger-looking number, but for unit fractions the inequality flips - star must be less than 7.
How to solve
Strategy Look for a Pattern — Both fractions have numerator 1, so the comparison depends only on the denominators: the unit fraction with the smaller denominator is larger. That pattern instantly tells me 1/star greater than 1/7 exactly when star less than 7, and I can list those digits to count them.
1STEP 1
Use the unit-fraction size rule
Cut a whole into more equal parts and each part shrinks, so with numerator 1 the fraction is bigger when the denominator is smaller.
1/7 less than 1/★ ⟺ ★ less than 7
Sharing a pizza among fewer people gives each person a bigger slice, so smaller denominator = bigger unit fraction.
3.NF.A.3Look For A PatternBecause both fractions have numerator 1, the unit fraction with the smaller denominator is the larger one, so 1/7 < 1/star holds exactly when star cuts the whole into fewer parts than 7.
Why?
Both 1/7 and 1/star name one equal piece of the same single whole, so which fraction is larger depends only on how many equal pieces that one whole was cut into.
Why?
A unit fraction 1/n is one part of a whole cut into n equal parts with no gaps or overlaps, and those n parts are exactly the one whole put back together.
🧱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?
Cutting that same whole into more equal parts forces each part to be smaller, so a bigger denominator gives a smaller unit fraction.
Why?
The n equal parts of size 1/n stack up to make exactly one whole, so when there are more of them each part must be smaller to keep that total whole unchanged.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
List the digits 1 to 9 that are less than 7
We need star less than 7, with star chosen from 1 through 9. The digits less than 7 are 1, 2, 3, 4, 5, and 6.
★ ∈ {1, 2, 3, 4, 5, 6}
Only denominators smaller than 7 make a slice bigger than 1/7; 7 itself ties and 8, 9 are smaller.
3.NF.A.3Make A Systematic List3STEP 3
Count the valid values
Count the digits in the list 1, 2, 3, 4, 5, 6.
6
There are exactly six whole numbers between 1 and 6, so six choices work.
3.NF.A.3Make A Systematic ListAnswer
6
Spot-check the boundaries: 1/6 greater than 1/7 (true, 6 counts) and 1/7 less than 1/7 is false (7 excluded) and 1/8 less than 1/7 so 1/7 less than 1/8 is false (8 excluded). Only 1-6 qualify, giving 6 - consistent with the answer.
Takeaway
For 1-over-something fractions, fewer parts means bigger pieces - so just count the denominators smaller than 7!
- Use the unit-fraction size rule
- List the digits 1 to 9 that are less than 7
- Count the valid values
Where next?
Another one like thissuggested
Step back and solidifysuggested