Problem
Multiply the two unit fractions on the left by multiplying the denominators: 5 times 12 is 60. So the left side becomes one-sixtieth.
Multiplying two unit fractions just multiplies the bottom numbers — a fifth of a twelfth is one out of sixty equal pieces, which a 5th grader can picture.
5.NF.B.4Solve An Easier Related ProblemThe right side becomes one over (8 times the box), another unit fraction.
The same denominator-times-denominator rule turns the right side into a single unit fraction, so now both sides have a 1 on top.
5.NF.B.4Solve An Easier Related ProblemFor unit fractions a bigger denominator means a smaller fraction, so 8 times the box must stay under 60.
Cut a whole into more pieces and each piece is smaller; so the fraction with the bigger denominator is the smaller fraction. That flips the comparison into one about the denominators.
4.NF.A.2Solve An Easier Related ProblemBetween two unit fractions, the one with the smaller denominator is the larger.
A unit fraction is one piece of a whole cut into as many equal pieces as the denominator says, so cutting the whole less finely leaves each piece bigger.
Comparing one piece against one piece is therefore just comparing how finely each whole was cut.
The single piece taken from each side pairs straight across, so the side cut into fewer pieces holds the bigger one.
8 x 7 = 56 clears 60 but 8 x 8 = 64 does not, so the largest that works is 7.
Testing the two numbers right at the edge confirms exactly where 'works' turns into 'too big', so we are sure 7 is the largest.
3.NF.A.3Guess And CheckMultiply each side first, then remember: for unit fractions, a smaller bottom number means a bigger fraction.