Problem
Fraction of a quantity by multiplying
There are 30 apples. The number of pears is the number of apples plus an extra amount equal to 5/6 of the apples. We need the total number of pears.
Fractions
Your answer
How to solve
Strategy Identify Subproblems — The phrase "more than the apples by 5/6 of the apples" packs two ideas: the base 30 apples and an extra part that is 5/6 of 30. Splitting it into subproblems (find the extra part, then add it on) keeps the work clear. Thinking of "5/6 of a quantity" the same way you would "1/6 of a quantity" (an easier related idea) shows it is just a multiplication.
1STEP 1
See the total as a base plus an extra part
Pears = (the 30 apples) + (an extra 5/6 of the apples). So I will first find the extra part, then add it to 30.
pears = 30 + 5/6 × 30
Breaking 'more than X by Y' into 'X and then Y added' is the natural way a 5th grader reads a comparison sentence.
4.OA.A.3Identify Subproblems2STEP 2
Find 5/6 of 30
Finding a fraction of a number is multiplication. One sixth of 30 is 5, so five sixths of 30 is 5 copies of that: 25.
5/6 × 30 = (5 × 30)/6 = 150/6 = 25
Splitting 30 into 6 equal groups of 5 makes 'five sixths' easy: take 5 of those groups.
5.NF.B.4Solve An Easier Related Problem3STEP 3
Add the extra onto the base
Adding the extra 25 onto the base 30 gives 55, the same as 30 taken one and five-sixths times.
30 + 25 = 55 = 30×(1+5/6)
Combining the whole part and the fractional part is exactly the distributive rule, just read in everyday words.
5.NF.B.6Identify SubproblemsThe pears are 30 plus 5/6 of 30, which is the same as 30 taken one and five-sixths times.
Why?
The pear count is the apple count together with the extra part, and there is nothing else in that total.
🧱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?
One whole copy of 30 plus five-sixths of a copy is one-and-five-sixths copies, because a copy may be split into parts and the parts gathered back up.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
Answer
55 pears
55 pears is more than 30 apples, which matches 'the pears are more than the apples.' The extra 25 is a bit less than 30, which fits 5/6 (a little under one whole). Units stay as 'count of fruit' throughout.
Takeaway
"5/6 of a number" just means multiply by 5/6 — find the extra part, then add it on.
- See the total as a base plus an extra part
- Find 5/6 of 30
- Add the extra onto the base
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested