Problem
98 = 2 x 7 x 7, so a numerator shares a factor with it only by being a multiple of 2 or 7.
Breaking a number into its prime factors is a 4th-grade factoring skill; it tells us the only 'dangerous' factors are 2 and 7.
4.OA.B.4Solve An Easier Related ProblemThe numerators go from 1 to 97, so there are 97 fractions in all. This is the whole group we will count from.
Listing the numerators 1, 2, 3, ..., 97 in order makes it clear there are exactly 97 of them.
4.OA.B.4Make A Systematic ListMultiples of 2 up to 96 number 96 divided by 2 = 48, and every one of those fractions simplifies.
Multiples of 2 are easy to count by skip-counting, and each one shares the factor 2 with 98.
4.OA.B.4Change Focus Count The ComplementMultiples of 7 up to 91 number 91 divided by 7 = 13, and they share the factor 7.
Skip-counting by 7 lists every multiple of 7 below 98, and each shares the factor 7 with 98.
4.OA.B.4Change Focus Count The ComplementMultiples of 14 land in both lists, and there are 84 divided by 14 = 6 of them to give back.
A number counted in both lists should only be removed once; listing the multiples of 14 shows there are 6 of them.
4.OA.B.4Make A Systematic ListSo the numerators sharing a factor with 98 come to 48 + 13 - 6 = 55.
Adding the two groups and taking out the shared part once is the same careful counting kids use to avoid counting anyone twice.
4.NF.A.1Change Focus Count The ComplementThe numerators that share a factor with 98 number 48 plus 13 minus 6.
98 is built out of 2 and 7 only, so a numerator shares a factor with it exactly when it is a multiple of 2 or a multiple of 7.
A shared factor bigger than 1 has to be made from the very primes 98 is built out of, so it carries a 2 or a 7.
Numerators that are multiples of both 2 and 7 land in each list, so they were counted twice and one of those counts has to be given back.
Counting a group means matching each member to exactly one tally, so a member that picked up two tallies must return one.
The fractions in lowest terms are the ones left over: 97 total minus the 55 that simplify. 97 - 55 = 42.
Whatever is not in the 'can be simplified' pile must be in lowest terms, so a single subtraction finishes the count.
4.NF.A.1Change Focus Count The ComplementIf the top and bottom share a factor, the fraction can shrink - so count the shrinkable ones and subtract: 97 - 55 = 42 stay in lowest terms.