Smaller divisor makes a bigger quotient

4.NBT.B.6 · take · grade 4

Archetype: Build the Largest or Smallest Value from Digit Cards · step in a 7-type progression

Using the number cards $5$ , $4$ , $1$ , $8$ , $2$ each exactly once, form a (3-digit number) $\div$ (2-digit number) division that makes the quotient as large as possible, then compute it.

Show solution

Understand

Use the five digit cards 5, 4, 1, 8, 2 once each to make a 3-digit number divided by a 2-digit number. Arrange them so the quotient is as large as possible, then carry out the division.

Givens

Digit cards available: 5, 4, 1, 8, 2 (each used exactly once).
The expression is (3-digit number) divided by (2-digit number).
Goal: make the quotient (the result of the division) as large as possible.

Unknowns

The arrangement of digits giving the largest quotient, and that quotient (with remainder).

Constraints

Each of the five digits is used exactly once.
Three digits form the dividend, the other two form the divisor.

Plan

#6 Guess and Check · also uses: #8 Analyze the Units#2 Make a Systematic List

A quotient grows when the dividend is big and the divisor is small. So make the divisor the smallest possible 2-digit number and the dividend the biggest 3-digit number from the remaining cards, then check the division.

Execute

#6 Guess and Check 4.NBT.B.6

Dividing by a smaller number makes the quotient larger. The smallest 2-digit number from these cards uses the two smallest digits, 1 and 2, with 1 in the tens place: 12.

\text{smallest divisor} = 12

Splitting something into fewer, smaller-size groups means each share is bigger, so a small divisor lifts the quotient.

#6 Guess and Check 4.NBT.A.2

The remaining cards are 8, 5, 4. The largest 3-digit number puts the biggest digit first: 854.

\text{largest dividend} = 854

Place the biggest digit in the hundreds spot to make the number as large as it can be.

#6 Guess and Check 4.NBT.B.6

Compute 854 divided by 12. Since 12 x 71 = 852, the quotient is 71 with a remainder of 2.

854 \div 12 = 71 \text{ remainder } 2 \quad (12 \times 71 = 852,\; 854 - 852 = 2)

Find how many 12s fit in 854; 71 of them reach 852, leaving 2 left over.

Answer: 854 / 12 = 71 remainder 2 (quotient 71)

Review

Compare with the next divisor option 14: 852 / 14 is about 60, much smaller than 71. Any divisor bigger than 12 lowers the quotient, and 854 is the biggest dividend left, so 854 / 12 truly gives the largest quotient. Check: 12 x 71 + 2 = 854.

Make a systematic list (tool 2): the only small divisors are 12, 14, 15, 18, 21...; pairing each with the largest leftover 3-digit number and dividing shows 854 / 12 = 71 beats them all, confirming the answer.

Standards · min grade 4

4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends — Dividing 854 by 12 to get quotient 71 and remainder 2.
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols — Building the largest 3-digit dividend and smallest 2-digit divisor by place value.

💡 Biggest top number over smallest bottom number makes the biggest answer - then just divide, which is Grade 4 work you know!