Sensim Math · Depth 한국어

3-2 · Division

Recover the dividend from quotient and remainder

3.OA.B.63.OA.A.4 · take · grade 3

Archetype: Divisibility and Remainder Reasoning · step in a 8-type progression

▶ Practice — 12 problems

When a certain number is divided by 8, the quotient is 9 and the remainder is 1. Find the quotient and the remainder when this number is divided by 5.

Show solution

Understand

A number divided by 8 gives quotient 9 and remainder 1. We first recover that number, then divide it by 5 to find the new quotient and remainder.

Givens
  • The number divided by 8 has quotient 9 and remainder 1.
  • We then divide the same number by 5.
Unknowns
  • The quotient and remainder when the number is divided by 5.
Constraints
  • Remainders must be less than the divisor (less than 8 here, less than 5 in the second division).
  • The number is a whole number.

Plan

#11 Work Backwards · also uses: #6 Guess and Check

Rebuild the original number from quotient times divisor plus remainder, then carry out the new division by 5.

Execute

#11 Work Backwards 3.OA.B.6
The number equals 8 times the quotient 9, plus the remainder 1.
8×9+1=72+1=738 \times 9 + 1 = 72 + 1 = 73
Multiplying the quotient by the divisor and adding the remainder undoes the division.
#6 Guess and Check 3.OA.C.7
Now divide 73 by 5. Five goes into 73 fourteen times, using 70, and 3 are left over.
73÷5=14⋯373 \div 5 = 14 \cdots 3
Since 5 times 14 is 70 and 73 minus 70 is 3, the quotient is 14 and the remainder 3 (less than 5).
Answer: Quotient 14, remainder 3

Review

Check: 5 times 14 plus 3 equals 73, the recovered number, and the remainder 3 is less than the divisor 5, so the answer is valid.

Convert to an equation (tool 13): the number N satisfies N = 8 times 9 + 1 = 73, then solve N = 5q + r with 0 less than or equal to r less than 5, giving q = 14, r = 3.

Standards · min grade 3

  • 3.OA.B.6 Understand division as an unknown-factor problem — Rebuilding the number 73 from quotient 9 and remainder 1.
  • 3.OA.C.7 Fluently multiply and divide within 100 — Dividing 73 by 5 to get quotient 14 and remainder 3.
💡 This only needs Grade 3 division: multiply back to find the number, then divide it again!