Problem
Apply conditions step by step, then count
I need to count how many fractions are exactly the same size as 6/13 while also having a denominator strictly between 50 and 100 and a numerator strictly between 20 and 80.
Fractions
Your answer
How to solve
Strategy Make a Systematic List — Every fraction equal to 6/13 is built by multiplying both 6 and 13 by the same whole number k, so I can list the equal fractions one k at a time and check each one against the denominator and numerator conditions. Listing keeps me from missing or double-counting any fraction.
1STEP 1
Write the form of every equal fraction
Every fraction equal to 6/13 has numerator 6k and denominator 13k for a whole number k.
6/13=(6 × k)/(13 × k)
Fourth graders know equivalent fractions come from multiplying top and bottom by the same number, so this captures every fraction the same size as 6/13.
4.NF.A.1Make A Systematic ListEvery fraction equal to 6/13 has the form 6k over 13k for some whole number k.
Why?
Multiplying the numerator and the denominator by the same whole number leaves the value alone, so 6k over 13k really is 6/13.
Why?
A number written over itself is one, and multiplying by one changes nothing.
🧱Multiplicative identity (one)One group of a number is just that number — multiplying by one changes nothing.
Why?
Nothing else can work, because 6 and 13 share no factor, so 6/13 is already in lowest terms and any equal fraction must be it scaled up whole.
Why?
Building the fraction any other way would leave different factors on the top and the bottom than 6 and 13 carry.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
2STEP 2
Use the denominator condition to limit k
The denominator 13k has to land between 50 and 100, which happens for k = 4, 5, 6, 7.
13×4=52, 13×5=65, 13×6=78, 13×7=91
Counting multiples of 13 in order makes it easy to see exactly which ones land in the 50-to-100 window.
4.OA.B.4Make A Systematic List3STEP 3
Check the numerator condition for those k
Their numerators 24, 30, 36 and 42 all sit between 20 and 80, so all four pass.
6×4=24, 6×5=30, 6×6=36, 6×7=42
Testing each candidate's numerator against the range is a quick guess-and-check that confirms which fractions truly satisfy every rule.
4.NBT.B.5Guess And Check4STEP 4
Count the fractions that pass everything
The fractions that satisfy all three conditions are 24/52, 30/65, 36/78, and 42/91. That is 4 fractions.
24/52, 30/65, 36/78, 42/91
Once the list is complete, counting the survivors gives the answer directly.
4.NF.A.1Make A Systematic ListAnswer
4 fractions
Each listed fraction reduces back to 6/13 (for example 24/52 = 6/13), and every denominator (52, 65, 78, 91) sits between 50 and 100 while every numerator (24, 30, 36, 42) sits between 20 and 80. The answer is a small whole number, which fits a 'how many' counting question.
Takeaway
Equivalent fractions come from one multiplier k, so list 6k/13k in order and just count the ones that fit both ranges.
- Write the form of every equal fraction
- Use the denominator condition to limit k
- Check the numerator condition for those k
- Count the fractions that pass everything
Where next?
Another one like thissuggested
Step back and solidifysuggested