Practice · Split off a multiple of the divisor to read the quotient · Sensim Math

Variant 1 easy

A four-digit whole number has thousands digit $x$ , hundreds digit $y$ , tens digit $z$ , and ones digit $3$ . When this number is divided by $2$ , what is the quotient? Write it using $x$ , $y$ and $z$ . (The quotient is a whole number.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A four-digit number has digits $x$ , $y$ , $z$ , $3$ (thousands, hundreds, tens, ones). Divide it by $2$ and write the whole-number quotient as an expression in $x$ , $y$ and $z$ .

Givens

Thousands digit $x$ , hundreds digit $y$ , tens digit $z$ , ones digit $3$ .
The number is divided by $2$ .
The quotient is a whole number.

Unknowns

The quotient, written in terms of $x$ , $y$ and $z$ .

Constraints

A remainder must be smaller than the divisor $2$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Write the number by place value, then split it into a part that is a multiple of $2$ plus a small leftover. The multiple of $2$ divided by $2$ is the quotient; the leftover is the remainder.

3 · Execute4 carry out the plan

1Write the number by place value

#13 Convert to Algebra 7.EE.A.2

A thousands digit is worth

1000

each, a hundreds digit

100

each and a tens digit

10

each, so the number is

1000x+100y+10z+3

This is how

2343

means

2000+300+40+3

.

2Break the ones digit into a multiple of 2 plus a leftover

#7 Identify Subproblems 7.EE.A.2

1000x

,

100y

and

10z

are already multiples of

2

. Only the

3

is not, so split it as

2+1

.

1000x+100y+10z+3=(1000x+100y+10z+2)+1

Peel off the part that

2

goes into evenly.

3Factor out 2

#13 Convert to Algebra 7.EE.A.1

Take

2

out of every term of the first group.

1000x+100y+10z+2=2(500x+50y+5z+1)

This is the distributive property run backwards.

4Read off the quotient

#7 Identify Subproblems 7.EE.A.2

So the number is

2(500x+50y+5z+1)+1

. The leftover

1

is less than

2

, so it is the remainder, and the quotient is

500x+50y+5z+1

.

1000x+100y+10z+3=2(500x+50y+5z+1)+1

Dividend

=

divisor

\times

quotient

+

remainder, with remainder smaller than the divisor.

Answer:

500x+50y+5z+1

4 · Reviewdoes it hold up?

Try $x=2$ , $y=3$ , $z=4$ : the number is $2343$ , and $2343=2\times1171+1$ , so the quotient is $1171$ . The formula gives $500\times2+50\times3+5\times4+1=1171$ . It matches.

Another way: Solve an easier case first: $1003\div2$ has quotient $501=500\times1+1$ , and each extra $100$ in the hundreds place adds $50$ and each extra $10$ in the tens place adds $5$ to the quotient, which suggests $500x+50y+5z+1$ directly.

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the number as a multiple of $2$ plus a remainder.
7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Factoring $2$ out of $1000x+100y+10z+2$.

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

Variant 2 easy

A three-digit whole number has hundreds digit $x$ , tens digit $y$ , and ones digit $5$ . When this number is divided by $2$ , what is the quotient? Write it using $x$ and $y$ . (The quotient is a whole number.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A three-digit number has digits $x$ , $y$ , $5$ (hundreds, tens, ones). Divide it by $2$ and write the whole-number quotient as an expression in $x$ and $y$ .

Givens

Hundreds digit $x$ , tens digit $y$ , ones digit $5$ .
The number is divided by $2$ .
The quotient is a whole number.

Unknowns

The quotient, written in terms of $x$ and $y$ .

Constraints

A remainder must be smaller than the divisor $2$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Write the number by place value, then split it into a part that is a multiple of $2$ plus a small leftover. The multiple of $2$ divided by $2$ is the quotient; the leftover is the remainder.

3 · Execute4 carry out the plan

1Write the number by place value

#13 Convert to Algebra 7.EE.A.2

A hundreds digit is worth

100

each and a tens digit

10

each, so the number is

100x+10y+5

This is how

345

means

300+40+5

.

2Break the ones digit into a multiple of 2 plus a leftover

#7 Identify Subproblems 7.EE.A.2

100x

and

10y

are already multiples of

2

. Only the

5

is not, so split it as

4+1

.

100x+10y+5=(100x+10y+4)+1

Peel off the part that

2

goes into evenly.

3Factor out 2

#13 Convert to Algebra 7.EE.A.1

Take

2

out of every term of the first group.

100x+10y+4=2(50x+5y+2)

This is the distributive property run backwards.

4Read off the quotient

#7 Identify Subproblems 7.EE.A.2

So the number is

2(50x+5y+2)+1

. The leftover

1

is less than

2

, so it is the remainder, and the quotient is

50x+5y+2

.

100x+10y+5=2(50x+5y+2)+1

Dividend

=

divisor

\times

quotient

+

remainder, with remainder smaller than the divisor.

Answer:

50x+5y+2

4 · Reviewdoes it hold up?

Try $x=3$ , $y=4$ : the number is $345$ , and $345=2\times172+1$ , so the quotient is $172$ . The formula gives $50\times3+5\times4+2=172$ . It matches.

Another way: Solve an easier case first: $105\div2$ has quotient $52=50\times1+2$ , and each extra $10$ in the tens place adds $5$ to the quotient, which suggests $50x+5y+2$ directly.

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the number as a multiple of $2$ plus a remainder.
7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Factoring $2$ out of $100x+10y+4$.

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

Variant 3 easy

A four-digit whole number has thousands digit $x$ , hundreds digit $y$ , tens digit $z$ , and ones digit $6$ . When this number is divided by $5$ , what is the quotient? Write it using $x$ , $y$ and $z$ . (The quotient is a whole number.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A four-digit number has digits $x$ , $y$ , $z$ , $6$ (thousands, hundreds, tens, ones). Divide it by $5$ and write the whole-number quotient as an expression in $x$ , $y$ and $z$ .

Givens

Thousands digit $x$ , hundreds digit $y$ , tens digit $z$ , ones digit $6$ .
The number is divided by $5$ .
The quotient is a whole number.

Unknowns

The quotient, written in terms of $x$ , $y$ and $z$ .

Constraints

A remainder must be smaller than the divisor $5$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

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.

3 · Execute4 carry out the plan

1Write the number by place value

#13 Convert to Algebra 7.EE.A.2

A thousands digit is worth

1000

each, a hundreds digit

100

each and a tens digit

10

each, so the number is

1000x+100y+10z+6

This is how

2346

means

2000+300+40+6

.

2Break the ones digit into a multiple of 5 plus a leftover

#7 Identify Subproblems 7.EE.A.2

1000x

,

100y

and

10z

are already multiples of

5

. Only the

6

is not, so split it as

5+1

.

1000x+100y+10z+6=(1000x+100y+10z+5)+1

Peel off the part that

5

goes into evenly.

3Factor out 5

#13 Convert to Algebra 7.EE.A.1

Take

5

out of every term of the first group.

1000x+100y+10z+5=5(200x+20y+2z+1)

This is the distributive property run backwards.

4Read off the quotient

#7 Identify Subproblems 7.EE.A.2

So the number is

5(200x+20y+2z+1)+1

. The leftover

1

is less than

5

, so it is the remainder, and the quotient is

200x+20y+2z+1

.

1000x+100y+10z+6=5(200x+20y+2z+1)+1

Dividend

=

divisor

\times

quotient

+

remainder, with remainder smaller than the divisor.

Answer:

200x+20y+2z+1

4 · Reviewdoes it hold up?

Try $x=2$ , $y=3$ , $z=4$ : the number is $2346$ , and $2346=5\times469+1$ , so the quotient is $469$ . The formula gives $200\times2+20\times3+2\times4+1=469$ . It matches.

Another way: Solve an easier case first: $1006\div5$ has quotient $201=200\times1+1$ , and each extra $100$ in the hundreds place adds $20$ and each extra $10$ in the tens place adds $2$ to the quotient, which suggests $200x+20y+2z+1$ directly.

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the number as a multiple of $5$ plus a remainder.
7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Factoring $5$ out of $1000x+100y+10z+5$.

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

Variant 4 easy

A three-digit whole number has hundreds digit $x$ , tens digit $y$ , and ones digit $6$ . When this number is divided by $5$ , what is the quotient? Write it using $x$ and $y$ . (The quotient is a whole number.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A three-digit number has digits $x$ , $y$ , $6$ (hundreds, tens, ones). Divide it by $5$ and write the whole-number quotient as an expression in $x$ and $y$ .

Givens

Hundreds digit $x$ , tens digit $y$ , ones digit $6$ .
The number is divided by $5$ .
The quotient is a whole number.

Unknowns

The quotient, written in terms of $x$ and $y$ .

Constraints

A remainder must be smaller than the divisor $5$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

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.

3 · Execute4 carry out the plan

1Write the number by place value

#13 Convert to Algebra 7.EE.A.2

A hundreds digit is worth

100

each and a tens digit

10

each, so the number is

100x+10y+6

This is how

346

means

300+40+6

.

2Break the ones digit into a multiple of 5 plus a leftover

#7 Identify Subproblems 7.EE.A.2

100x

and

10y

are already multiples of

5

. Only the

6

is not, so split it as

5+1

.

100x+10y+6=(100x+10y+5)+1

Peel off the part that

5

goes into evenly.

3Factor out 5

#13 Convert to Algebra 7.EE.A.1

Take

5

out of every term of the first group.

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

This is the distributive property run backwards.

4Read off the quotient

#7 Identify Subproblems 7.EE.A.2

So the number is

5(20x+2y+1)+1

. The leftover

1

is less than

5

, so it is the remainder, and the quotient is

20x+2y+1

.

100x+10y+6=5(20x+2y+1)+1

Dividend

=

divisor

\times

quotient

+

remainder, with remainder smaller than the divisor.

Answer:

20x+2y+1

4 · Reviewdoes it hold up?

Try $x=3$ , $y=4$ : the number is $346$ , and $346=5\times69+1$ , so the quotient is $69$ . The formula gives $20\times3+2\times4+1=69$ . It matches.

Another way: Solve an easier case first: $106\div5$ has quotient $21=20\times1+1$ , and each extra $10$ in the tens place adds $2$ to the quotient, which suggests $20x+2y+1$ directly.

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the number as a multiple of $5$ plus a remainder.
7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Factoring $5$ out of $100x+10y+5$.

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

Variant 5 medium

A three-digit whole number has hundreds digit $x$ , tens digit $y$ , and ones digit $7$ . When this number is divided by $2$ , what is the quotient? Write it using $x$ and $y$ . (The quotient is a whole number.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A three-digit number has digits $x$ , $y$ , $7$ (hundreds, tens, ones). Divide it by $2$ and write the whole-number quotient as an expression in $x$ and $y$ .

Givens

Hundreds digit $x$ , tens digit $y$ , ones digit $7$ .
The number is divided by $2$ .
The quotient is a whole number.

Unknowns

The quotient, written in terms of $x$ and $y$ .

Constraints

A remainder must be smaller than the divisor $2$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Write the number by place value, then split it into a part that is a multiple of $2$ plus a small leftover. The multiple of $2$ divided by $2$ is the quotient; the leftover is the remainder.

3 · Execute4 carry out the plan

1Write the number by place value

#13 Convert to Algebra 7.EE.A.2

A hundreds digit is worth

100

each and a tens digit

10

each, so the number is

100x+10y+7

This is how

347

means

300+40+7

.

2Break the ones digit into a multiple of 2 plus a leftover

#7 Identify Subproblems 7.EE.A.2

100x

and

10y

are already multiples of

2

. Only the

7

is not, so split it as

6+1

.

100x+10y+7=(100x+10y+6)+1

Peel off the part that

2

goes into evenly.

3Factor out 2

#13 Convert to Algebra 7.EE.A.1

Take

2

out of every term of the first group.

100x+10y+6=2(50x+5y+3)

This is the distributive property run backwards.

4Read off the quotient

#7 Identify Subproblems 7.EE.A.2

So the number is

2(50x+5y+3)+1

. The leftover

1

is less than

2

, so it is the remainder, and the quotient is

50x+5y+3

.

100x+10y+7=2(50x+5y+3)+1

Dividend

=

divisor

\times

quotient

+

remainder, with remainder smaller than the divisor.

Answer:

50x+5y+3

4 · Reviewdoes it hold up?

Try $x=3$ , $y=4$ : the number is $347$ , and $347=2\times173+1$ , so the quotient is $173$ . The formula gives $50\times3+5\times4+3=173$ . It matches.

Another way: Solve an easier case first: $107\div2$ has quotient $53=50\times1+3$ , and each extra $10$ in the tens place adds $5$ to the quotient, which suggests $50x+5y+3$ directly.

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the number as a multiple of $2$ plus a remainder.
7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Factoring $2$ out of $100x+10y+6$.

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

Variant 6 medium

A three-digit whole number has hundreds digit $x$ , tens digit $y$ , and ones digit $7$ . When this number is divided by $5$ , what is the quotient? Write it using $x$ and $y$ . (The quotient is a whole number.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

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 $y$ .

Givens

Hundreds digit $x$ , tens digit $y$ , ones digit $7$ .
The number is divided by $5$ .
The quotient is a whole number.

Unknowns

The quotient, written in terms of $x$ and $y$ .

Constraints

A remainder must be smaller than the divisor $5$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

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.

3 · Execute4 carry out the plan

1Write the number by place value

#13 Convert to Algebra 7.EE.A.2

A hundreds digit is worth

100

each and a tens digit

10

each, so the number is

100x+10y+7

This is how

347

means

300+40+7

.

2Break the ones digit into a multiple of 5 plus a leftover

#7 Identify Subproblems 7.EE.A.2

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

Peel off the part that

5

goes into evenly.

3Factor out 5

#13 Convert to Algebra 7.EE.A.1

Take

5

out of every term of the first group.

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

This is the distributive property run backwards.

4Read off the quotient

#7 Identify Subproblems 7.EE.A.2

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

Dividend

=

divisor

\times

quotient

+

remainder, with remainder smaller than the divisor.

Answer:

20x+2y+1

4 · Reviewdoes it hold up?

Try $x=3$ , $y=4$ : the number is $347$ , and $347=5\times69+2$ , so the quotient is $69$ . The formula gives $20\times3+2\times4+1=69$ . It matches.

Another way: Solve an easier case first: $107\div5$ has quotient $21=20\times1+1$ , and each extra $10$ in the tens place adds $2$ to the quotient, which suggests $20x+2y+1$ directly.

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the number as a multiple of $5$ plus a remainder.
7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Factoring $5$ out of $100x+10y+5$.

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

Variant 7 medium

A four-digit whole number has thousands digit $x$ , hundreds digit $y$ , tens digit $z$ , and ones digit $7$ . When this number is divided by $5$ , what is the quotient? Write it using $x$ , $y$ and $z$ . (The quotient is a whole number.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A four-digit number has digits $x$ , $y$ , $z$ , $7$ (thousands, hundreds, tens, ones). Divide it by $5$ and write the whole-number quotient as an expression in $x$ , $y$ and $z$ .

Givens

Thousands digit $x$ , hundreds digit $y$ , tens digit $z$ , ones digit $7$ .
The number is divided by $5$ .
The quotient is a whole number.

Unknowns

The quotient, written in terms of $x$ , $y$ and $z$ .

Constraints

A remainder must be smaller than the divisor $5$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

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.

3 · Execute4 carry out the plan

1Write the number by place value

#13 Convert to Algebra 7.EE.A.2

A thousands digit is worth

1000

each, a hundreds digit

100

each and a tens digit

10

each, so the number is

1000x+100y+10z+7

This is how

2347

means

2000+300+40+7

.

2Break the ones digit into a multiple of 5 plus a leftover

#7 Identify Subproblems 7.EE.A.2

1000x

,

100y

and

10z

are already multiples of

5

. Only the

7

is not, so split it as

5+2

.

1000x+100y+10z+7=(1000x+100y+10z+5)+2

Peel off the part that

5

goes into evenly.

3Factor out 5

#13 Convert to Algebra 7.EE.A.1

Take

5

out of every term of the first group.

1000x+100y+10z+5=5(200x+20y+2z+1)

This is the distributive property run backwards.

4Read off the quotient

#7 Identify Subproblems 7.EE.A.2

So the number is

5(200x+20y+2z+1)+2

. The leftover

2

is less than

5

, so it is the remainder, and the quotient is

200x+20y+2z+1

.

1000x+100y+10z+7=5(200x+20y+2z+1)+2

Dividend

=

divisor

\times

quotient

+

remainder, with remainder smaller than the divisor.

Answer:

200x+20y+2z+1

4 · Reviewdoes it hold up?

Try $x=2$ , $y=3$ , $z=4$ : the number is $2347$ , and $2347=5\times469+2$ , so the quotient is $469$ . The formula gives $200\times2+20\times3+2\times4+1=469$ . It matches.

Another way: Solve an easier case first: $1007\div5$ has quotient $201=200\times1+1$ , and each extra $100$ in the hundreds place adds $20$ and each extra $10$ in the tens place adds $2$ to the quotient, which suggests $200x+20y+2z+1$ directly.

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the number as a multiple of $5$ plus a remainder.
7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Factoring $5$ out of $1000x+100y+10z+5$.

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

Variant 8 medium

A four-digit whole number has thousands digit $x$ , hundreds digit $y$ , tens digit $z$ , and ones digit $8$ . When this number is divided by $5$ , what is the quotient? Write it using $x$ , $y$ and $z$ . (The quotient is a whole number.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A four-digit number has digits $x$ , $y$ , $z$ , $8$ (thousands, hundreds, tens, ones). Divide it by $5$ and write the whole-number quotient as an expression in $x$ , $y$ and $z$ .

Givens

Thousands digit $x$ , hundreds digit $y$ , tens digit $z$ , ones digit $8$ .
The number is divided by $5$ .
The quotient is a whole number.

Unknowns

The quotient, written in terms of $x$ , $y$ and $z$ .

Constraints

A remainder must be smaller than the divisor $5$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

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.

3 · Execute4 carry out the plan

1Write the number by place value

#13 Convert to Algebra 7.EE.A.2

A thousands digit is worth

1000

each, a hundreds digit

100

each and a tens digit

10

each, so the number is

1000x+100y+10z+8

This is how

2348

means

2000+300+40+8

.

2Break the ones digit into a multiple of 5 plus a leftover

#7 Identify Subproblems 7.EE.A.2

1000x

,

100y

and

10z

are already multiples of

5

. Only the

8

is not, so split it as

5+3

.

1000x+100y+10z+8=(1000x+100y+10z+5)+3

Peel off the part that

5

goes into evenly.

3Factor out 5

#13 Convert to Algebra 7.EE.A.1

Take

5

out of every term of the first group.

1000x+100y+10z+5=5(200x+20y+2z+1)

This is the distributive property run backwards.

4Read off the quotient

#7 Identify Subproblems 7.EE.A.2

So the number is

5(200x+20y+2z+1)+3

. The leftover

3

is less than

5

, so it is the remainder, and the quotient is

200x+20y+2z+1

.

1000x+100y+10z+8=5(200x+20y+2z+1)+3

Dividend

=

divisor

\times

quotient

+

remainder, with remainder smaller than the divisor.

Answer:

200x+20y+2z+1

4 · Reviewdoes it hold up?

Try $x=2$ , $y=3$ , $z=4$ : the number is $2348$ , and $2348=5\times469+3$ , so the quotient is $469$ . The formula gives $200\times2+20\times3+2\times4+1=469$ . It matches.

Another way: Solve an easier case first: $1008\div5$ has quotient $201=200\times1+1$ , and each extra $100$ in the hundreds place adds $20$ and each extra $10$ in the tens place adds $2$ to the quotient, which suggests $200x+20y+2z+1$ directly.

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the number as a multiple of $5$ plus a remainder.
7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Factoring $5$ out of $1000x+100y+10z+5$.

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

Variant 9 hard

A three-digit whole number has hundreds digit $x$ , tens digit $y$ , and ones digit $8$ . When this number is divided by $5$ , what is the quotient? Write it using $x$ and $y$ . (The quotient is a whole number.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A three-digit number has digits $x$ , $y$ , $8$ (hundreds, tens, ones). Divide it by $5$ and write the whole-number quotient as an expression in $x$ and $y$ .

Givens

Hundreds digit $x$ , tens digit $y$ , ones digit $8$ .
The number is divided by $5$ .
The quotient is a whole number.

Unknowns

The quotient, written in terms of $x$ and $y$ .

Constraints

A remainder must be smaller than the divisor $5$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

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.

3 · Execute4 carry out the plan

1Write the number by place value

#13 Convert to Algebra 7.EE.A.2

A hundreds digit is worth

100

each and a tens digit

10

each, so the number is

100x+10y+8

This is how

348

means

300+40+8

.

2Break the ones digit into a multiple of 5 plus a leftover

#7 Identify Subproblems 7.EE.A.2

100x

and

10y

are already multiples of

5

. Only the

8

is not, so split it as

5+3

.

100x+10y+8=(100x+10y+5)+3

Peel off the part that

5

goes into evenly.

3Factor out 5

#13 Convert to Algebra 7.EE.A.1

Take

5

out of every term of the first group.

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

This is the distributive property run backwards.

4Read off the quotient

#7 Identify Subproblems 7.EE.A.2

So the number is

5(20x+2y+1)+3

. The leftover

3

is less than

5

, so it is the remainder, and the quotient is

20x+2y+1

.

100x+10y+8=5(20x+2y+1)+3

Dividend

=

divisor

\times

quotient

+

remainder, with remainder smaller than the divisor.

Answer:

20x+2y+1

4 · Reviewdoes it hold up?

Try $x=3$ , $y=4$ : the number is $348$ , and $348=5\times69+3$ , so the quotient is $69$ . The formula gives $20\times3+2\times4+1=69$ . It matches.

Another way: Solve an easier case first: $108\div5$ has quotient $21=20\times1+1$ , and each extra $10$ in the tens place adds $2$ to the quotient, which suggests $20x+2y+1$ directly.

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the number as a multiple of $5$ plus a remainder.
7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Factoring $5$ out of $100x+10y+5$.

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

Variant 10 hard

A three-digit whole number has hundreds digit $x$ , tens digit $y$ , and ones digit $9$ . When this number is divided by $2$ , what is the quotient? Write it using $x$ and $y$ . (The quotient is a whole number.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A three-digit number has digits $x$ , $y$ , $9$ (hundreds, tens, ones). Divide it by $2$ and write the whole-number quotient as an expression in $x$ and $y$ .

Givens

Hundreds digit $x$ , tens digit $y$ , ones digit $9$ .
The number is divided by $2$ .
The quotient is a whole number.

Unknowns

The quotient, written in terms of $x$ and $y$ .

Constraints

A remainder must be smaller than the divisor $2$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Write the number by place value, then split it into a part that is a multiple of $2$ plus a small leftover. The multiple of $2$ divided by $2$ is the quotient; the leftover is the remainder.

3 · Execute4 carry out the plan

1Write the number by place value

#13 Convert to Algebra 7.EE.A.2

A hundreds digit is worth

100

each and a tens digit

10

each, so the number is

100x+10y+9

This is how

349

means

300+40+9

.

2Break the ones digit into a multiple of 2 plus a leftover

#7 Identify Subproblems 7.EE.A.2

100x

and

10y

are already multiples of

2

. Only the

9

is not, so split it as

8+1

.

100x+10y+9=(100x+10y+8)+1

Peel off the part that

2

goes into evenly.

3Factor out 2

#13 Convert to Algebra 7.EE.A.1

Take

2

out of every term of the first group.

100x+10y+8=2(50x+5y+4)

This is the distributive property run backwards.

4Read off the quotient

#7 Identify Subproblems 7.EE.A.2

So the number is

2(50x+5y+4)+1

. The leftover

1

is less than

2

, so it is the remainder, and the quotient is

50x+5y+4

.

100x+10y+9=2(50x+5y+4)+1

Dividend

=

divisor

\times

quotient

+

remainder, with remainder smaller than the divisor.

Answer:

50x+5y+4

4 · Reviewdoes it hold up?

Try $x=3$ , $y=4$ : the number is $349$ , and $349=2\times174+1$ , so the quotient is $174$ . The formula gives $50\times3+5\times4+4=174$ . It matches.

Another way: Solve an easier case first: $109\div2$ has quotient $54=50\times1+4$ , and each extra $10$ in the tens place adds $5$ to the quotient, which suggests $50x+5y+4$ directly.

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the number as a multiple of $2$ plus a remainder.
7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Factoring $2$ out of $100x+10y+8$.

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

Variant 11 hard

A four-digit whole number has thousands digit $x$ , hundreds digit $y$ , tens digit $z$ , and ones digit $9$ . When this number is divided by $2$ , what is the quotient? Write it using $x$ , $y$ and $z$ . (The quotient is a whole number.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A four-digit number has digits $x$ , $y$ , $z$ , $9$ (thousands, hundreds, tens, ones). Divide it by $2$ and write the whole-number quotient as an expression in $x$ , $y$ and $z$ .

Givens

Thousands digit $x$ , hundreds digit $y$ , tens digit $z$ , ones digit $9$ .
The number is divided by $2$ .
The quotient is a whole number.

Unknowns

The quotient, written in terms of $x$ , $y$ and $z$ .

Constraints

A remainder must be smaller than the divisor $2$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Write the number by place value, then split it into a part that is a multiple of $2$ plus a small leftover. The multiple of $2$ divided by $2$ is the quotient; the leftover is the remainder.

3 · Execute4 carry out the plan

1Write the number by place value

#13 Convert to Algebra 7.EE.A.2

A thousands digit is worth

1000

each, a hundreds digit

100

each and a tens digit

10

each, so the number is

1000x+100y+10z+9

This is how

2349

means

2000+300+40+9

.

2Break the ones digit into a multiple of 2 plus a leftover

#7 Identify Subproblems 7.EE.A.2

1000x

,

100y

and

10z

are already multiples of

2

. Only the

9

is not, so split it as

8+1

.

1000x+100y+10z+9=(1000x+100y+10z+8)+1

Peel off the part that

2

goes into evenly.

3Factor out 2

#13 Convert to Algebra 7.EE.A.1

Take

2

out of every term of the first group.

1000x+100y+10z+8=2(500x+50y+5z+4)

This is the distributive property run backwards.

4Read off the quotient

#7 Identify Subproblems 7.EE.A.2

So the number is

2(500x+50y+5z+4)+1

. The leftover

1

is less than

2

, so it is the remainder, and the quotient is

500x+50y+5z+4

.

1000x+100y+10z+9=2(500x+50y+5z+4)+1

Dividend

=

divisor

\times

quotient

+

remainder, with remainder smaller than the divisor.

Answer:

500x+50y+5z+4

4 · Reviewdoes it hold up?

Try $x=2$ , $y=3$ , $z=4$ : the number is $2349$ , and $2349=2\times1174+1$ , so the quotient is $1174$ . The formula gives $500\times2+50\times3+5\times4+4=1174$ . It matches.

Another way: Solve an easier case first: $1009\div2$ has quotient $504=500\times1+4$ , and each extra $100$ in the hundreds place adds $50$ and each extra $10$ in the tens place adds $5$ to the quotient, which suggests $500x+50y+5z+4$ directly.

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the number as a multiple of $2$ plus a remainder.
7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Factoring $2$ out of $1000x+100y+10z+8$.

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

Variant 12 hard

A three-digit whole number has hundreds digit $x$ , tens digit $y$ , and ones digit $9$ . When this number is divided by $5$ , what is the quotient? Write it using $x$ and $y$ . (The quotient is a whole number.)

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A three-digit number has digits $x$ , $y$ , $9$ (hundreds, tens, ones). Divide it by $5$ and write the whole-number quotient as an expression in $x$ and $y$ .

Givens

Hundreds digit $x$ , tens digit $y$ , ones digit $9$ .
The number is divided by $5$ .
The quotient is a whole number.

Unknowns

The quotient, written in terms of $x$ and $y$ .

Constraints

A remainder must be smaller than the divisor $5$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

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.

3 · Execute4 carry out the plan

1Write the number by place value

#13 Convert to Algebra 7.EE.A.2

A hundreds digit is worth

100

each and a tens digit

10

each, so the number is

100x+10y+9

This is how

349

means

300+40+9

.

2Break the ones digit into a multiple of 5 plus a leftover

#7 Identify Subproblems 7.EE.A.2

100x

and

10y

are already multiples of

5

. Only the

9

is not, so split it as

5+4

.

100x+10y+9=(100x+10y+5)+4

Peel off the part that

5

goes into evenly.

3Factor out 5

#13 Convert to Algebra 7.EE.A.1

Take

5

out of every term of the first group.

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

This is the distributive property run backwards.

4Read off the quotient

#7 Identify Subproblems 7.EE.A.2

So the number is

5(20x+2y+1)+4

. The leftover

4

is less than

5

, so it is the remainder, and the quotient is

20x+2y+1

.

100x+10y+9=5(20x+2y+1)+4

Dividend

=

divisor

\times

quotient

+

remainder, with remainder smaller than the divisor.

Answer:

20x+2y+1

4 · Reviewdoes it hold up?

Try $x=3$ , $y=4$ : the number is $349$ , and $349=5\times69+4$ , so the quotient is $69$ . The formula gives $20\times3+2\times4+1=69$ . It matches.

Another way: Solve an easier case first: $109\div5$ has quotient $21=20\times1+1$ , and each extra $10$ in the tens place adds $2$ to the quotient, which suggests $20x+2y+1$ directly.

Standardsmin grade 7

7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Rewriting the number as a multiple of $5$ plus a remainder.
7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Factoring $5$ out of $100x+10y+5$.

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

Split off a multiple of the divisor to read the quotient · 12 practice problems

Generated variants — 12

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