Operations & Word Problems

Depth · one concept from first principles · step 2 of 4

Split off a multiple of 5 to read the quotient

A three-digit number has digits x, y, 7 (hundreds, tens, ones). Divide it by 5 and write the whole-number quotient as an expression in x and y.
Expressions
Your answer
How to solve
Strategy Convert to Algebra — Write the number by place value, then split it into a part that is a multiple of 5 plus a small leftover. The multiple of 5 divided by 5 is the quotient; the leftover is the remainder.
1STEP 1

Write the number by place value

A hundreds digit is worth 100 each and a tens digit 10 each, so the number is

100x+10y+7
2STEP 2

Break the ones digit into a multiple of 5 plus a leftover

100x and 10y are already multiples of 5. Only the 7 is not, so split it as 5+2.

100x+10y+7=(100x+10y+5)+2
3STEP 3

Factor out 5

Take 5 out of every term of the first group.

100x+10y+5=5(20x+2y+1)
4STEP 4

Read off the quotient

So the number is 5(20x+2y+1)+2. The leftover 2 is less than 5, so it is the remainder, and the quotient is 20x+2y+1.

100x+10y+7=5(20x+2y+1)+2
Answer
20x+2y+1
Try x=3, y=4: the number is 347, and 347=5×69+2, so the quotient is 69. The formula gives 20×3+2×4+1=69. It matches.
Takeaway

Write a number by its place values, peel off a multiple of the divisor, and the quotient appears.

  • Write the number by place value
  • Break the ones digit into a multiple of 5 plus a leftover
  • Factor out 5
  • Read off the quotient
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