Practice · Divisor times quotient plus remainder gives the number back · Sensim Math

Variant 1 easy

A number was divided by $4.3$ and the quotient taken only as far as its whole-number part, giving a quotient of $7$ and a remainder of $1.8$ . If the same number is divided by $6.6$ in the same way, what is the remainder?

Your answer

Show solution

1 · Understandwhat's really being asked

Some number over 4.3 gives 7 remainder 1.8. Divided by 6.6 the same way, we want the new remainder.

Givens

Dividing by 4.3 gives quotient 7 and remainder 1.8.
The same number is then divided by 6.6.
In both cases the quotient is taken only as far as its whole-number part.

Unknowns

The remainder from the second division.

Constraints

The number itself is never stated.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#8 Analyze the Units

Nothing can be done with the second divisor until the number is known. Divisor times quotient plus remainder rebuilds it exactly, and then the second division is just a division.

3 · Execute3 carry out the plan

1Rebuild the number

#11 Work Backwards 6.NS.B.2

Multiply the divisor by the quotient and add the remainder.

4.3 \times 7 + 1.8 = 31.9

The number is 31.9.

2See how many 6.6s fit

#7 Identify Subproblems 6.NS.B.3

4 of them come to 26.4, and one more would overshoot.

6.6 \times 4 = 26.4

The whole-number quotient is 4.

3Take that off the number

#8 Analyze the Units 6.NS.B.3

What is left over is the remainder.

31.9 - 26.4 = 5.5

The remainder is 5.5.

Answer: 5.5

4 · Reviewdoes it hold up?

The remainder 5.5 is smaller than the divisor 6.6, as every remainder must be -- otherwise another whole 6.6 would have fitted.

Another way: Rebuilding from the second division instead, $6.6 \times 4 + 5.5 = 31.9$ , gives back the same number, which is the check that both divisions describe it.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing and subtracting decimals.
6.NS.B.2 Fluently divide multi-digit numbers — Using divisor times quotient plus remainder.

💡Takeaway. A quotient and a remainder together say exactly what the number was. You can always build it back.

Variant 2 easy

A number was divided by $4.8$ and the quotient taken only as far as its whole-number part, giving a quotient of $6$ and a remainder of $2.7$ . If the same number is divided by $7.1$ in the same way, what is the remainder?

Your answer

Show solution

1 · Understandwhat's really being asked

Some number over 4.8 gives 6 remainder 2.7. Divided by 7.1 the same way, we want the new remainder.

Givens

Dividing by 4.8 gives quotient 6 and remainder 2.7.
The same number is then divided by 7.1.
In both cases the quotient is taken only as far as its whole-number part.

Unknowns

The remainder from the second division.

Constraints

The number itself is never stated.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#8 Analyze the Units

Nothing can be done with the second divisor until the number is known. Divisor times quotient plus remainder rebuilds it exactly, and then the second division is just a division.

3 · Execute3 carry out the plan

1Rebuild the number

#11 Work Backwards 6.NS.B.2

Multiply the divisor by the quotient and add the remainder.

4.8 \times 6 + 2.7 = 31.5

The number is 31.5.

2See how many 7.1s fit

#7 Identify Subproblems 6.NS.B.3

4 of them come to 28.4, and one more would overshoot.

7.1 \times 4 = 28.4

The whole-number quotient is 4.

3Take that off the number

#8 Analyze the Units 6.NS.B.3

What is left over is the remainder.

31.5 - 28.4 = 3.1

The remainder is 3.1.

Answer: 3.1

4 · Reviewdoes it hold up?

The remainder 3.1 is smaller than the divisor 7.1, as every remainder must be -- otherwise another whole 7.1 would have fitted.

Another way: Rebuilding from the second division instead, $7.1 \times 4 + 3.1 = 31.5$ , gives back the same number, which is the check that both divisions describe it.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing and subtracting decimals.
6.NS.B.2 Fluently divide multi-digit numbers — Using divisor times quotient plus remainder.

💡Takeaway. A quotient and a remainder together say exactly what the number was. You can always build it back.

Variant 3 easy

A number was divided by $5.2$ and the quotient taken only as far as its whole-number part, giving a quotient of $4$ and a remainder of $2.1$ . If the same number is divided by $7.3$ in the same way, what is the remainder?

Your answer

Show solution

1 · Understandwhat's really being asked

Some number over 5.2 gives 4 remainder 2.1. Divided by 7.3 the same way, we want the new remainder.

Givens

Dividing by 5.2 gives quotient 4 and remainder 2.1.
The same number is then divided by 7.3.
In both cases the quotient is taken only as far as its whole-number part.

Unknowns

The remainder from the second division.

Constraints

The number itself is never stated.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#8 Analyze the Units

Nothing can be done with the second divisor until the number is known. Divisor times quotient plus remainder rebuilds it exactly, and then the second division is just a division.

3 · Execute3 carry out the plan

1Rebuild the number

#11 Work Backwards 6.NS.B.2

Multiply the divisor by the quotient and add the remainder.

5.2 \times 4 + 2.1 = 22.9

The number is 22.9.

2See how many 7.3s fit

#7 Identify Subproblems 6.NS.B.3

3 of them come to 21.9, and one more would overshoot.

7.3 \times 3 = 21.9

The whole-number quotient is 3.

3Take that off the number

#8 Analyze the Units 6.NS.B.3

What is left over is the remainder.

22.9 - 21.9 = 1

The remainder is 1.

Answer: 1

4 · Reviewdoes it hold up?

The remainder 1 is smaller than the divisor 7.3, as every remainder must be -- otherwise another whole 7.3 would have fitted.

Another way: Rebuilding from the second division instead, $7.3 \times 3 + 1 = 22.9$ , gives back the same number, which is the check that both divisions describe it.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing and subtracting decimals.
6.NS.B.2 Fluently divide multi-digit numbers — Using divisor times quotient plus remainder.

💡Takeaway. A quotient and a remainder together say exactly what the number was. You can always build it back.

Variant 4 easy

A number was divided by $7.4$ and the quotient taken only as far as its whole-number part, giving a quotient of $7$ and a remainder of $4.6$ . If the same number is divided by $8.3$ in the same way, what is the remainder?

Your answer

Show solution

1 · Understandwhat's really being asked

Some number over 7.4 gives 7 remainder 4.6. Divided by 8.3 the same way, we want the new remainder.

Givens

Dividing by 7.4 gives quotient 7 and remainder 4.6.
The same number is then divided by 8.3.
In both cases the quotient is taken only as far as its whole-number part.

Unknowns

The remainder from the second division.

Constraints

The number itself is never stated.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#8 Analyze the Units

Nothing can be done with the second divisor until the number is known. Divisor times quotient plus remainder rebuilds it exactly, and then the second division is just a division.

3 · Execute3 carry out the plan

1Rebuild the number

#11 Work Backwards 6.NS.B.2

Multiply the divisor by the quotient and add the remainder.

7.4 \times 7 + 4.6 = 56.4

The number is 56.4.

2See how many 8.3s fit

#7 Identify Subproblems 6.NS.B.3

6 of them come to 49.8, and one more would overshoot.

8.3 \times 6 = 49.8

The whole-number quotient is 6.

3Take that off the number

#8 Analyze the Units 6.NS.B.3

What is left over is the remainder.

56.4 - 49.8 = 6.6

The remainder is 6.6.

Answer: 6.6

4 · Reviewdoes it hold up?

The remainder 6.6 is smaller than the divisor 8.3, as every remainder must be -- otherwise another whole 8.3 would have fitted.

Another way: Rebuilding from the second division instead, $8.3 \times 6 + 6.6 = 56.4$ , gives back the same number, which is the check that both divisions describe it.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing and subtracting decimals.
6.NS.B.2 Fluently divide multi-digit numbers — Using divisor times quotient plus remainder.

💡Takeaway. A quotient and a remainder together say exactly what the number was. You can always build it back.

Variant 5 medium

A number was divided by $3.7$ and the quotient taken only as far as its whole-number part, giving a quotient of $8$ and a remainder of $2.5$ . If the same number is divided by $8.4$ in the same way, what is the remainder?

Your answer

Show solution

1 · Understandwhat's really being asked

Some number over 3.7 gives 8 remainder 2.5. Divided by 8.4 the same way, we want the new remainder.

Givens

Dividing by 3.7 gives quotient 8 and remainder 2.5.
The same number is then divided by 8.4.
In both cases the quotient is taken only as far as its whole-number part.

Unknowns

The remainder from the second division.

Constraints

The number itself is never stated.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#8 Analyze the Units

Nothing can be done with the second divisor until the number is known. Divisor times quotient plus remainder rebuilds it exactly, and then the second division is just a division.

3 · Execute3 carry out the plan

1Rebuild the number

#11 Work Backwards 6.NS.B.2

Multiply the divisor by the quotient and add the remainder.

3.7 \times 8 + 2.5 = 32.1

The number is 32.1.

2See how many 8.4s fit

#7 Identify Subproblems 6.NS.B.3

3 of them come to 25.2, and one more would overshoot.

8.4 \times 3 = 25.2

The whole-number quotient is 3.

3Take that off the number

#8 Analyze the Units 6.NS.B.3

What is left over is the remainder.

32.1 - 25.2 = 6.9

The remainder is 6.9.

Answer: 6.9

4 · Reviewdoes it hold up?

The remainder 6.9 is smaller than the divisor 8.4, as every remainder must be -- otherwise another whole 8.4 would have fitted.

Another way: Rebuilding from the second division instead, $8.4 \times 3 + 6.9 = 32.1$ , gives back the same number, which is the check that both divisions describe it.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing and subtracting decimals.
6.NS.B.2 Fluently divide multi-digit numbers — Using divisor times quotient plus remainder.

💡Takeaway. A quotient and a remainder together say exactly what the number was. You can always build it back.

Variant 6 medium

A number was divided by $8.5$ and the quotient taken only as far as its whole-number part, giving a quotient of $5$ and a remainder of $3.9$ . If the same number is divided by $6.2$ in the same way, what is the remainder?

Your answer

Show solution

1 · Understandwhat's really being asked

Some number over 8.5 gives 5 remainder 3.9. Divided by 6.2 the same way, we want the new remainder.

Givens

Dividing by 8.5 gives quotient 5 and remainder 3.9.
The same number is then divided by 6.2.
In both cases the quotient is taken only as far as its whole-number part.

Unknowns

The remainder from the second division.

Constraints

The number itself is never stated.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#8 Analyze the Units

Nothing can be done with the second divisor until the number is known. Divisor times quotient plus remainder rebuilds it exactly, and then the second division is just a division.

3 · Execute3 carry out the plan

1Rebuild the number

#11 Work Backwards 6.NS.B.2

Multiply the divisor by the quotient and add the remainder.

8.5 \times 5 + 3.9 = 46.4

The number is 46.4.

2See how many 6.2s fit

#7 Identify Subproblems 6.NS.B.3

7 of them come to 43.4, and one more would overshoot.

6.2 \times 7 = 43.4

The whole-number quotient is 7.

3Take that off the number

#8 Analyze the Units 6.NS.B.3

What is left over is the remainder.

46.4 - 43.4 = 3

The remainder is 3.

Answer: 3

4 · Reviewdoes it hold up?

The remainder 3 is smaller than the divisor 6.2, as every remainder must be -- otherwise another whole 6.2 would have fitted.

Another way: Rebuilding from the second division instead, $6.2 \times 7 + 3 = 46.4$ , gives back the same number, which is the check that both divisions describe it.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing and subtracting decimals.
6.NS.B.2 Fluently divide multi-digit numbers — Using divisor times quotient plus remainder.

💡Takeaway. A quotient and a remainder together say exactly what the number was. You can always build it back.

Variant 7 medium

A number was divided by $6.9$ and the quotient taken only as far as its whole-number part, giving a quotient of $4$ and a remainder of $3.3$ . If the same number is divided by $8.8$ in the same way, what is the remainder?

Your answer

Show solution

1 · Understandwhat's really being asked

Some number over 6.9 gives 4 remainder 3.3. Divided by 8.8 the same way, we want the new remainder.

Givens

Dividing by 6.9 gives quotient 4 and remainder 3.3.
The same number is then divided by 8.8.
In both cases the quotient is taken only as far as its whole-number part.

Unknowns

The remainder from the second division.

Constraints

The number itself is never stated.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#8 Analyze the Units

Nothing can be done with the second divisor until the number is known. Divisor times quotient plus remainder rebuilds it exactly, and then the second division is just a division.

3 · Execute3 carry out the plan

1Rebuild the number

#11 Work Backwards 6.NS.B.2

Multiply the divisor by the quotient and add the remainder.

6.9 \times 4 + 3.3 = 30.9

The number is 30.9.

2See how many 8.8s fit

#7 Identify Subproblems 6.NS.B.3

3 of them come to 26.4, and one more would overshoot.

8.8 \times 3 = 26.4

The whole-number quotient is 3.

3Take that off the number

#8 Analyze the Units 6.NS.B.3

What is left over is the remainder.

30.9 - 26.4 = 4.5

The remainder is 4.5.

Answer: 4.5

4 · Reviewdoes it hold up?

The remainder 4.5 is smaller than the divisor 8.8, as every remainder must be -- otherwise another whole 8.8 would have fitted.

Another way: Rebuilding from the second division instead, $8.8 \times 3 + 4.5 = 30.9$ , gives back the same number, which is the check that both divisions describe it.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing and subtracting decimals.
6.NS.B.2 Fluently divide multi-digit numbers — Using divisor times quotient plus remainder.

💡Takeaway. A quotient and a remainder together say exactly what the number was. You can always build it back.

Variant 8 medium

A number was divided by $5.6$ and the quotient taken only as far as its whole-number part, giving a quotient of $9$ and a remainder of $4.1$ . If the same number is divided by $7.7$ in the same way, what is the remainder?

Your answer

Show solution

1 · Understandwhat's really being asked

Some number over 5.6 gives 9 remainder 4.1. Divided by 7.7 the same way, we want the new remainder.

Givens

Dividing by 5.6 gives quotient 9 and remainder 4.1.
The same number is then divided by 7.7.
In both cases the quotient is taken only as far as its whole-number part.

Unknowns

The remainder from the second division.

Constraints

The number itself is never stated.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#8 Analyze the Units

Nothing can be done with the second divisor until the number is known. Divisor times quotient plus remainder rebuilds it exactly, and then the second division is just a division.

3 · Execute3 carry out the plan

1Rebuild the number

#11 Work Backwards 6.NS.B.2

Multiply the divisor by the quotient and add the remainder.

5.6 \times 9 + 4.1 = 54.5

The number is 54.5.

2See how many 7.7s fit

#7 Identify Subproblems 6.NS.B.3

7 of them come to 53.9, and one more would overshoot.

7.7 \times 7 = 53.9

The whole-number quotient is 7.

3Take that off the number

#8 Analyze the Units 6.NS.B.3

What is left over is the remainder.

54.5 - 53.9 = 0.6

The remainder is 0.6.

Answer: 0.6

4 · Reviewdoes it hold up?

The remainder 0.6 is smaller than the divisor 7.7, as every remainder must be -- otherwise another whole 7.7 would have fitted.

Another way: Rebuilding from the second division instead, $7.7 \times 7 + 0.6 = 54.5$ , gives back the same number, which is the check that both divisions describe it.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing and subtracting decimals.
6.NS.B.2 Fluently divide multi-digit numbers — Using divisor times quotient plus remainder.

💡Takeaway. A quotient and a remainder together say exactly what the number was. You can always build it back.

Variant 9 hard

A number was divided by $9.1$ and the quotient taken only as far as its whole-number part, giving a quotient of $3$ and a remainder of $4.7$ . If the same number is divided by $5.8$ in the same way, what is the remainder?

Your answer

Variant 10 hard

A number was divided by $7.8$ and the quotient taken only as far as its whole-number part, giving a quotient of $6$ and a remainder of $3.5$ . If the same number is divided by $9.4$ in the same way, what is the remainder?

Your answer

Show solution

1 · Understandwhat's really being asked

Some number over 7.8 gives 6 remainder 3.5. Divided by 9.4 the same way, we want the new remainder.

Givens

Dividing by 7.8 gives quotient 6 and remainder 3.5.
The same number is then divided by 9.4.
In both cases the quotient is taken only as far as its whole-number part.

Unknowns

The remainder from the second division.

Constraints

The number itself is never stated.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#8 Analyze the Units

Nothing can be done with the second divisor until the number is known. Divisor times quotient plus remainder rebuilds it exactly, and then the second division is just a division.

3 · Execute3 carry out the plan

1Rebuild the number

#11 Work Backwards 6.NS.B.2

Multiply the divisor by the quotient and add the remainder.

7.8 \times 6 + 3.5 = 50.3

The number is 50.3.

2See how many 9.4s fit

#7 Identify Subproblems 6.NS.B.3

5 of them come to 47, and one more would overshoot.

9.4 \times 5 = 47

The whole-number quotient is 5.

3Take that off the number

#8 Analyze the Units 6.NS.B.3

What is left over is the remainder.

50.3 - 47 = 3.3

The remainder is 3.3.

Answer: 3.3

4 · Reviewdoes it hold up?

The remainder 3.3 is smaller than the divisor 9.4, as every remainder must be -- otherwise another whole 9.4 would have fitted.

Another way: Rebuilding from the second division instead, $9.4 \times 5 + 3.3 = 50.3$ , gives back the same number, which is the check that both divisions describe it.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing and subtracting decimals.
6.NS.B.2 Fluently divide multi-digit numbers — Using divisor times quotient plus remainder.

💡Takeaway. A quotient and a remainder together say exactly what the number was. You can always build it back.

Variant 11 hard

A number was divided by $6.4$ and the quotient taken only as far as its whole-number part, giving a quotient of $5$ and a remainder of $1.2$ . If the same number is divided by $9.5$ in the same way, what is the remainder?

Your answer

Show solution

1 · Understandwhat's really being asked

Some number over 6.4 gives 5 remainder 1.2. Divided by 9.5 the same way, we want the new remainder.

Givens

Dividing by 6.4 gives quotient 5 and remainder 1.2.
The same number is then divided by 9.5.
In both cases the quotient is taken only as far as its whole-number part.

Unknowns

The remainder from the second division.

Constraints

The number itself is never stated.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#8 Analyze the Units

Nothing can be done with the second divisor until the number is known. Divisor times quotient plus remainder rebuilds it exactly, and then the second division is just a division.

3 · Execute3 carry out the plan

1Rebuild the number

#11 Work Backwards 6.NS.B.2

Multiply the divisor by the quotient and add the remainder.

6.4 \times 5 + 1.2 = 33.2

The number is 33.2.

2See how many 9.5s fit

#7 Identify Subproblems 6.NS.B.3

3 of them come to 28.5, and one more would overshoot.

9.5 \times 3 = 28.5

The whole-number quotient is 3.

3Take that off the number

#8 Analyze the Units 6.NS.B.3

What is left over is the remainder.

33.2 - 28.5 = 4.7

The remainder is 4.7.

Answer: 4.7

4 · Reviewdoes it hold up?

The remainder 4.7 is smaller than the divisor 9.5, as every remainder must be -- otherwise another whole 9.5 would have fitted.

Another way: Rebuilding from the second division instead, $9.5 \times 3 + 4.7 = 33.2$ , gives back the same number, which is the check that both divisions describe it.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing and subtracting decimals.
6.NS.B.2 Fluently divide multi-digit numbers — Using divisor times quotient plus remainder.

💡Takeaway. A quotient and a remainder together say exactly what the number was. You can always build it back.

Variant 12 hard

A number was divided by $9.6$ and the quotient taken only as far as its whole-number part, giving a quotient of $2$ and a remainder of $5.3$ . If the same number is divided by $6.7$ in the same way, what is the remainder?

Your answer

Show solution

1 · Understandwhat's really being asked

Some number over 9.6 gives 2 remainder 5.3. Divided by 6.7 the same way, we want the new remainder.

Givens

Dividing by 9.6 gives quotient 2 and remainder 5.3.
The same number is then divided by 6.7.
In both cases the quotient is taken only as far as its whole-number part.

Unknowns

The remainder from the second division.

Constraints

The number itself is never stated.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#8 Analyze the Units

Nothing can be done with the second divisor until the number is known. Divisor times quotient plus remainder rebuilds it exactly, and then the second division is just a division.

3 · Execute3 carry out the plan

1Rebuild the number

#11 Work Backwards 6.NS.B.2

Multiply the divisor by the quotient and add the remainder.

9.6 \times 2 + 5.3 = 24.5

The number is 24.5.

2See how many 6.7s fit

#7 Identify Subproblems 6.NS.B.3

3 of them come to 20.1, and one more would overshoot.

6.7 \times 3 = 20.1

The whole-number quotient is 3.

3Take that off the number

#8 Analyze the Units 6.NS.B.3

What is left over is the remainder.

24.5 - 20.1 = 4.4

The remainder is 4.4.

Answer: 4.4

4 · Reviewdoes it hold up?

The remainder 4.4 is smaller than the divisor 6.7, as every remainder must be -- otherwise another whole 6.7 would have fitted.

Another way: Rebuilding from the second division instead, $6.7 \times 3 + 4.4 = 24.5$ , gives back the same number, which is the check that both divisions describe it.

Standardsmin grade 6

6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing and subtracting decimals.
6.NS.B.2 Fluently divide multi-digit numbers — Using divisor times quotient plus remainder.

💡Takeaway. A quotient and a remainder together say exactly what the number was. You can always build it back.

Divisor times quotient plus remainder gives the number back · 12 practice problems

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