Practice · Divisor of dividend minus remainder, within bounds · Sensim Math

Variant 1 easy

Ivy's father is $41$ years old. When her father's age is divided by Ivy's age, the remainder is $8$ . Ivy's older sister, Alice, is $15$ years old. How old is Ivy? (Ages are whole numbers of years.)

Your answer

Show solution

1 · Understandwhat's really being asked

Dividing $41$ by Ivy's age leaves a remainder of $8$ , and Ivy is younger than her $15$ -year-old sister. Find Ivy's age.

Givens

Ivy's father is $41$ years old.
$41$ divided by Ivy's age leaves a remainder of $8$ .
Ivy's older sister Alice is $15$ years old.

Unknowns

Ivy's age.

Constraints

A remainder is always smaller than the divisor, so Ivy is older than $8$ .
Ivy is younger than her older sister, so she is younger than $15$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #2 Make a Systematic List#3 Eliminate Possibilities

Undo the division: taking the remainder away leaves an exact multiple of Ivy's age. List the divisors of that number, then keep only the one that fits between the two bounds.

3 · Execute3 carry out the plan

1Take the remainder away

#11 Work Backwards 4.OA.B.4

If Ivy is

x

years old and the quotient is

Q

, then

41 = x \times Q + 8

. Removing the

8

left over leaves a number that

x

divides exactly.

41 - 8 = 33 = x \times Q

Ivy's age must be a divisor of

33

.

2List the divisors of 33

#2 Make a Systematic List 4.OA.B.4

Write

33

as a product in every way.

33 = 1 \times 33 = 3 \times 11

The divisors are

1

,

3

,

11

,

33

.

3Apply both bounds

#3 Eliminate Possibilities 7.EE.B.4

The divisor must be bigger than the remainder

8

, and Ivy is younger than

15

. Only one divisor sits in that range.

8 < x < 15 \Rightarrow x = 11

Ivy is

11

.

Answer: 11 years old

4 · Reviewdoes it hold up?

Check: $11 \times 3 + 8 = 41$ , and $8 < 11$ , so $41$ divided by $11$ really leaves $8$ . Also $11 < 15$ .

Another way: Guess and check the ages $9$ to $14$ : dividing $41$ by each, only $11$ leaves a remainder of $8$ .

Standardsmin grade 7

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the divisors of $33$.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Using the bounds $8 < x < 15$ to pick the age.

💡Takeaway. Peel off the remainder and the divisor shows up as a factor -- then the bounds pick the right one.

Variant 2 easy

Ruby's father is $43$ years old. When her father's age is divided by Ruby's age, the remainder is $11$ . Ruby's older sister, Clara, is $23$ years old. How old is Ruby? (Ages are whole numbers of years.)

Your answer

Show solution

1 · Understandwhat's really being asked

Dividing $43$ by Ruby's age leaves a remainder of $11$ , and Ruby is younger than her $23$ -year-old sister. Find Ruby's age.

Givens

Ruby's father is $43$ years old.
$43$ divided by Ruby's age leaves a remainder of $11$ .
Ruby's older sister Clara is $23$ years old.

Unknowns

Ruby's age.

Constraints

A remainder is always smaller than the divisor, so Ruby is older than $11$ .
Ruby is younger than her older sister, so she is younger than $23$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #2 Make a Systematic List#3 Eliminate Possibilities

Undo the division: taking the remainder away leaves an exact multiple of Ruby's age. List the divisors of that number, then keep only the one that fits between the two bounds.

3 · Execute3 carry out the plan

1Take the remainder away

#11 Work Backwards 4.OA.B.4

If Ruby is

x

years old and the quotient is

Q

, then

43 = x \times Q + 11

. Removing the

11

left over leaves a number that

x

divides exactly.

43 - 11 = 32 = x \times Q

Ruby's age must be a divisor of

32

.

2List the divisors of 32

#2 Make a Systematic List 4.OA.B.4

Write

32

as a product in every way.

32 = 1 \times 32 = 2 \times 16 = 4 \times 8

The divisors are

1

,

2

,

4

,

8

,

16

,

32

.

3Apply both bounds

#3 Eliminate Possibilities 7.EE.B.4

The divisor must be bigger than the remainder

11

, and Ruby is younger than

23

. Only one divisor sits in that range.

11 < x < 23 \Rightarrow x = 16

Ruby is

16

.

Answer: 16 years old

4 · Reviewdoes it hold up?

Check: $16 \times 2 + 11 = 43$ , and $11 < 16$ , so $43$ divided by $16$ really leaves $11$ . Also $16 < 23$ .

Another way: Guess and check the ages $12$ to $22$ : dividing $43$ by each, only $16$ leaves a remainder of $11$ .

Standardsmin grade 7

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the divisors of $32$.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Using the bounds $11 < x < 23$ to pick the age.

💡Takeaway. Peel off the remainder and the divisor shows up as a factor -- then the bounds pick the right one.

Variant 3 easy

Zoe's father is $45$ years old. When her father's age is divided by Zoe's age, the remainder is $6$ . Zoe's older sister, Hannah, is $18$ years old. How old is Zoe? (Ages are whole numbers of years.)

Your answer

Show solution

1 · Understandwhat's really being asked

Dividing $45$ by Zoe's age leaves a remainder of $6$ , and Zoe is younger than her $18$ -year-old sister. Find Zoe's age.

Givens

Zoe's father is $45$ years old.
$45$ divided by Zoe's age leaves a remainder of $6$ .
Zoe's older sister Hannah is $18$ years old.

Unknowns

Zoe's age.

Constraints

A remainder is always smaller than the divisor, so Zoe is older than $6$ .
Zoe is younger than her older sister, so she is younger than $18$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #2 Make a Systematic List#3 Eliminate Possibilities

Undo the division: taking the remainder away leaves an exact multiple of Zoe's age. List the divisors of that number, then keep only the one that fits between the two bounds.

3 · Execute3 carry out the plan

1Take the remainder away

#11 Work Backwards 4.OA.B.4

If Zoe is

x

years old and the quotient is

Q

, then

45 = x \times Q + 6

. Removing the

6

left over leaves a number that

x

divides exactly.

45 - 6 = 39 = x \times Q

Zoe's age must be a divisor of

39

.

2List the divisors of 39

#2 Make a Systematic List 4.OA.B.4

Write

39

as a product in every way.

39 = 1 \times 39 = 3 \times 13

The divisors are

1

,

3

,

13

,

39

.

3Apply both bounds

#3 Eliminate Possibilities 7.EE.B.4

The divisor must be bigger than the remainder

6

, and Zoe is younger than

18

. Only one divisor sits in that range.

6 < x < 18 \Rightarrow x = 13

Zoe is

13

.

Answer: 13 years old

4 · Reviewdoes it hold up?

Check: $13 \times 3 + 6 = 45$ , and $6 < 13$ , so $45$ divided by $13$ really leaves $6$ . Also $13 < 18$ .

Another way: Guess and check the ages $7$ to $17$ : dividing $45$ by each, only $13$ leaves a remainder of $6$ .

Standardsmin grade 7

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the divisors of $39$.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Using the bounds $6 < x < 18$ to pick the age.

💡Takeaway. Peel off the remainder and the divisor shows up as a factor -- then the bounds pick the right one.

Variant 4 easy

Ava's father is $47$ years old. When her father's age is divided by Ava's age, the remainder is $9$ . Ava's older sister, Chloe, is $26$ years old. How old is Ava? (Ages are whole numbers of years.)

Your answer

Show solution

1 · Understandwhat's really being asked

Dividing $47$ by Ava's age leaves a remainder of $9$ , and Ava is younger than her $26$ -year-old sister. Find Ava's age.

Givens

Ava's father is $47$ years old.
$47$ divided by Ava's age leaves a remainder of $9$ .
Ava's older sister Chloe is $26$ years old.

Unknowns

Ava's age.

Constraints

A remainder is always smaller than the divisor, so Ava is older than $9$ .
Ava is younger than her older sister, so she is younger than $26$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #2 Make a Systematic List#3 Eliminate Possibilities

Undo the division: taking the remainder away leaves an exact multiple of Ava's age. List the divisors of that number, then keep only the one that fits between the two bounds.

3 · Execute3 carry out the plan

1Take the remainder away

#11 Work Backwards 4.OA.B.4

If Ava is

x

years old and the quotient is

Q

, then

47 = x \times Q + 9

. Removing the

9

left over leaves a number that

x

divides exactly.

47 - 9 = 38 = x \times Q

Ava's age must be a divisor of

38

.

2List the divisors of 38

#2 Make a Systematic List 4.OA.B.4

Write

38

as a product in every way.

38 = 1 \times 38 = 2 \times 19

The divisors are

1

,

2

,

19

,

38

.

3Apply both bounds

#3 Eliminate Possibilities 7.EE.B.4

The divisor must be bigger than the remainder

9

, and Ava is younger than

26

. Only one divisor sits in that range.

9 < x < 26 \Rightarrow x = 19

Ava is

19

.

Answer: 19 years old

4 · Reviewdoes it hold up?

Check: $19 \times 2 + 9 = 47$ , and $9 < 19$ , so $47$ divided by $19$ really leaves $9$ . Also $19 < 26$ .

Another way: Guess and check the ages $10$ to $25$ : dividing $47$ by each, only $19$ leaves a remainder of $9$ .

Standardsmin grade 7

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the divisors of $38$.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Using the bounds $9 < x < 26$ to pick the age.

💡Takeaway. Peel off the remainder and the divisor shows up as a factor -- then the bounds pick the right one.

Variant 5 medium

Lily's father is $48$ years old. When her father's age is divided by Lily's age, the remainder is $9$ . Lily's older sister, Grace, is $17$ years old. How old is Lily? (Ages are whole numbers of years.)

Your answer

Show solution

1 · Understandwhat's really being asked

Dividing $48$ by Lily's age leaves a remainder of $9$ , and Lily is younger than her $17$ -year-old sister. Find Lily's age.

Givens

Lily's father is $48$ years old.
$48$ divided by Lily's age leaves a remainder of $9$ .
Lily's older sister Grace is $17$ years old.

Unknowns

Lily's age.

Constraints

A remainder is always smaller than the divisor, so Lily is older than $9$ .
Lily is younger than her older sister, so she is younger than $17$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #2 Make a Systematic List#3 Eliminate Possibilities

Undo the division: taking the remainder away leaves an exact multiple of Lily's age. List the divisors of that number, then keep only the one that fits between the two bounds.

3 · Execute3 carry out the plan

1Take the remainder away

#11 Work Backwards 4.OA.B.4

If Lily is

x

years old and the quotient is

Q

, then

48 = x \times Q + 9

. Removing the

9

left over leaves a number that

x

divides exactly.

48 - 9 = 39 = x \times Q

Lily's age must be a divisor of

39

.

2List the divisors of 39

#2 Make a Systematic List 4.OA.B.4

Write

39

as a product in every way.

39 = 1 \times 39 = 3 \times 13

The divisors are

1

,

3

,

13

,

39

.

3Apply both bounds

#3 Eliminate Possibilities 7.EE.B.4

The divisor must be bigger than the remainder

9

, and Lily is younger than

17

. Only one divisor sits in that range.

9 < x < 17 \Rightarrow x = 13

Lily is

13

.

Answer: 13 years old

4 · Reviewdoes it hold up?

Check: $13 \times 3 + 9 = 48$ , and $9 < 13$ , so $48$ divided by $13$ really leaves $9$ . Also $13 < 17$ .

Another way: Guess and check the ages $10$ to $16$ : dividing $48$ by each, only $13$ leaves a remainder of $9$ .

Standardsmin grade 7

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the divisors of $39$.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Using the bounds $9 < x < 17$ to pick the age.

💡Takeaway. Peel off the remainder and the divisor shows up as a factor -- then the bounds pick the right one.

Variant 6 medium

Mia's father is $51$ years old. When her father's age is divided by Mia's age, the remainder is $13$ . Mia's older sister, Emma, is $22$ years old. How old is Mia? (Ages are whole numbers of years.)

Your answer

Show solution

1 · Understandwhat's really being asked

Dividing $51$ by Mia's age leaves a remainder of $13$ , and Mia is younger than her $22$ -year-old sister. Find Mia's age.

Givens

Mia's father is $51$ years old.
$51$ divided by Mia's age leaves a remainder of $13$ .
Mia's older sister Emma is $22$ years old.

Unknowns

Mia's age.

Constraints

A remainder is always smaller than the divisor, so Mia is older than $13$ .
Mia is younger than her older sister, so she is younger than $22$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #2 Make a Systematic List#3 Eliminate Possibilities

Undo the division: taking the remainder away leaves an exact multiple of Mia's age. List the divisors of that number, then keep only the one that fits between the two bounds.

3 · Execute3 carry out the plan

1Take the remainder away

#11 Work Backwards 4.OA.B.4

If Mia is

x

years old and the quotient is

Q

, then

51 = x \times Q + 13

. Removing the

13

left over leaves a number that

x

divides exactly.

51 - 13 = 38 = x \times Q

Mia's age must be a divisor of

38

.

2List the divisors of 38

#2 Make a Systematic List 4.OA.B.4

Write

38

as a product in every way.

38 = 1 \times 38 = 2 \times 19

The divisors are

1

,

2

,

19

,

38

.

3Apply both bounds

#3 Eliminate Possibilities 7.EE.B.4

The divisor must be bigger than the remainder

13

, and Mia is younger than

22

. Only one divisor sits in that range.

13 < x < 22 \Rightarrow x = 19

Mia is

19

.

Answer: 19 years old

4 · Reviewdoes it hold up?

Check: $19 \times 2 + 13 = 51$ , and $13 < 19$ , so $51$ divided by $19$ really leaves $13$ . Also $19 < 22$ .

Another way: Guess and check the ages $14$ to $21$ : dividing $51$ by each, only $19$ leaves a remainder of $13$ .

Standardsmin grade 7

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the divisors of $38$.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Using the bounds $13 < x < 22$ to pick the age.

💡Takeaway. Peel off the remainder and the divisor shows up as a factor -- then the bounds pick the right one.

Variant 7 medium

Nora's father is $53$ years old. When her father's age is divided by Nora's age, the remainder is $15$ . Nora's older sister, Ella, is $20$ years old. How old is Nora? (Ages are whole numbers of years.)

Your answer

Show solution

1 · Understandwhat's really being asked

Dividing $53$ by Nora's age leaves a remainder of $15$ , and Nora is younger than her $20$ -year-old sister. Find Nora's age.

Givens

Nora's father is $53$ years old.
$53$ divided by Nora's age leaves a remainder of $15$ .
Nora's older sister Ella is $20$ years old.

Unknowns

Nora's age.

Constraints

A remainder is always smaller than the divisor, so Nora is older than $15$ .
Nora is younger than her older sister, so she is younger than $20$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #2 Make a Systematic List#3 Eliminate Possibilities

Undo the division: taking the remainder away leaves an exact multiple of Nora's age. List the divisors of that number, then keep only the one that fits between the two bounds.

3 · Execute3 carry out the plan

1Take the remainder away

#11 Work Backwards 4.OA.B.4

If Nora is

x

years old and the quotient is

Q

, then

53 = x \times Q + 15

. Removing the

15

left over leaves a number that

x

divides exactly.

53 - 15 = 38 = x \times Q

Nora's age must be a divisor of

38

.

2List the divisors of 38

#2 Make a Systematic List 4.OA.B.4

Write

38

as a product in every way.

38 = 1 \times 38 = 2 \times 19

The divisors are

1

,

2

,

19

,

38

.

3Apply both bounds

#3 Eliminate Possibilities 7.EE.B.4

The divisor must be bigger than the remainder

15

, and Nora is younger than

20

. Only one divisor sits in that range.

15 < x < 20 \Rightarrow x = 19

Nora is

19

.

Answer: 19 years old

4 · Reviewdoes it hold up?

Check: $19 \times 2 + 15 = 53$ , and $15 < 19$ , so $53$ divided by $19$ really leaves $15$ . Also $19 < 20$ .

Another way: Guess and check the ages $16$ to $19$ : dividing $53$ by each, only $19$ leaves a remainder of $15$ .

Standardsmin grade 7

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the divisors of $38$.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Using the bounds $15 < x < 20$ to pick the age.

💡Takeaway. Peel off the remainder and the divisor shows up as a factor -- then the bounds pick the right one.

Variant 8 medium

Sofia's father is $54$ years old. When her father's age is divided by Sofia's age, the remainder is $14$ . Sofia's older sister, Leah, is $23$ years old. How old is Sofia? (Ages are whole numbers of years.)

Your answer

Show solution

1 · Understandwhat's really being asked

Dividing $54$ by Sofia's age leaves a remainder of $14$ , and Sofia is younger than her $23$ -year-old sister. Find Sofia's age.

Givens

Sofia's father is $54$ years old.
$54$ divided by Sofia's age leaves a remainder of $14$ .
Sofia's older sister Leah is $23$ years old.

Unknowns

Sofia's age.

Constraints

A remainder is always smaller than the divisor, so Sofia is older than $14$ .
Sofia is younger than her older sister, so she is younger than $23$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #2 Make a Systematic List#3 Eliminate Possibilities

Undo the division: taking the remainder away leaves an exact multiple of Sofia's age. List the divisors of that number, then keep only the one that fits between the two bounds.

3 · Execute3 carry out the plan

1Take the remainder away

#11 Work Backwards 4.OA.B.4

If Sofia is

x

years old and the quotient is

Q

, then

54 = x \times Q + 14

. Removing the

14

left over leaves a number that

x

divides exactly.

54 - 14 = 40 = x \times Q

Sofia's age must be a divisor of

40

.

2List the divisors of 40

#2 Make a Systematic List 4.OA.B.4

Write

40

as a product in every way.

40 = 1 \times 40 = 2 \times 20 = 4 \times 10 = 5 \times 8

The divisors are

1

,

2

,

4

,

5

,

8

,

10

,

20

,

40

.

3Apply both bounds

#3 Eliminate Possibilities 7.EE.B.4

The divisor must be bigger than the remainder

14

, and Sofia is younger than

23

. Only one divisor sits in that range.

14 < x < 23 \Rightarrow x = 20

Sofia is

20

.

Answer: 20 years old

4 · Reviewdoes it hold up?

Check: $20 \times 2 + 14 = 54$ , and $14 < 20$ , so $54$ divided by $20$ really leaves $14$ . Also $20 < 23$ .

Another way: Guess and check the ages $15$ to $22$ : dividing $54$ by each, only $20$ leaves a remainder of $14$ .

Standardsmin grade 7

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the divisors of $40$.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Using the bounds $14 < x < 23$ to pick the age.

💡Takeaway. Peel off the remainder and the divisor shows up as a factor -- then the bounds pick the right one.

Variant 9 hard

Lucy's father is $55$ years old. When her father's age is divided by Lucy's age, the remainder is $9$ . Lucy's older sister, Anna, is $25$ years old. How old is Lucy? (Ages are whole numbers of years.)

Your answer

Show solution

1 · Understandwhat's really being asked

Dividing $55$ by Lucy's age leaves a remainder of $9$ , and Lucy is younger than her $25$ -year-old sister. Find Lucy's age.

Givens

Lucy's father is $55$ years old.
$55$ divided by Lucy's age leaves a remainder of $9$ .
Lucy's older sister Anna is $25$ years old.

Unknowns

Lucy's age.

Constraints

A remainder is always smaller than the divisor, so Lucy is older than $9$ .
Lucy is younger than her older sister, so she is younger than $25$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #2 Make a Systematic List#3 Eliminate Possibilities

Undo the division: taking the remainder away leaves an exact multiple of Lucy's age. List the divisors of that number, then keep only the one that fits between the two bounds.

3 · Execute3 carry out the plan

1Take the remainder away

#11 Work Backwards 4.OA.B.4

If Lucy is

x

years old and the quotient is

Q

, then

55 = x \times Q + 9

. Removing the

9

left over leaves a number that

x

divides exactly.

55 - 9 = 46 = x \times Q

Lucy's age must be a divisor of

46

.

2List the divisors of 46

#2 Make a Systematic List 4.OA.B.4

Write

46

as a product in every way.

46 = 1 \times 46 = 2 \times 23

The divisors are

1

,

2

,

23

,

46

.

3Apply both bounds

#3 Eliminate Possibilities 7.EE.B.4

The divisor must be bigger than the remainder

9

, and Lucy is younger than

25

. Only one divisor sits in that range.

9 < x < 25 \Rightarrow x = 23

Lucy is

23

.

Answer: 23 years old

4 · Reviewdoes it hold up?

Check: $23 \times 2 + 9 = 55$ , and $9 < 23$ , so $55$ divided by $23$ really leaves $9$ . Also $23 < 25$ .

Another way: Guess and check the ages $10$ to $24$ : dividing $55$ by each, only $23$ leaves a remainder of $9$ .

Standardsmin grade 7

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the divisors of $46$.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Using the bounds $9 < x < 25$ to pick the age.

💡Takeaway. Peel off the remainder and the divisor shows up as a factor -- then the bounds pick the right one.

Variant 10 hard

Isla's father is $57$ years old. When her father's age is divided by Isla's age, the remainder is $6$ . Isla's older sister, Amy, is $22$ years old. How old is Isla? (Ages are whole numbers of years.)

Your answer

Show solution

1 · Understandwhat's really being asked

Dividing $57$ by Isla's age leaves a remainder of $6$ , and Isla is younger than her $22$ -year-old sister. Find Isla's age.

Givens

Isla's father is $57$ years old.
$57$ divided by Isla's age leaves a remainder of $6$ .
Isla's older sister Amy is $22$ years old.

Unknowns

Isla's age.

Constraints

A remainder is always smaller than the divisor, so Isla is older than $6$ .
Isla is younger than her older sister, so she is younger than $22$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #2 Make a Systematic List#3 Eliminate Possibilities

Undo the division: taking the remainder away leaves an exact multiple of Isla's age. List the divisors of that number, then keep only the one that fits between the two bounds.

3 · Execute3 carry out the plan

1Take the remainder away

#11 Work Backwards 4.OA.B.4

If Isla is

x

years old and the quotient is

Q

, then

57 = x \times Q + 6

. Removing the

6

left over leaves a number that

x

divides exactly.

57 - 6 = 51 = x \times Q

Isla's age must be a divisor of

51

.

2List the divisors of 51

#2 Make a Systematic List 4.OA.B.4

Write

51

as a product in every way.

51 = 1 \times 51 = 3 \times 17

The divisors are

1

,

3

,

17

,

51

.

3Apply both bounds

#3 Eliminate Possibilities 7.EE.B.4

The divisor must be bigger than the remainder

6

, and Isla is younger than

22

. Only one divisor sits in that range.

6 < x < 22 \Rightarrow x = 17

Isla is

17

.

Answer: 17 years old

4 · Reviewdoes it hold up?

Check: $17 \times 3 + 6 = 57$ , and $6 < 17$ , so $57$ divided by $17$ really leaves $6$ . Also $17 < 22$ .

Another way: Guess and check the ages $7$ to $21$ : dividing $57$ by each, only $17$ leaves a remainder of $6$ .

Standardsmin grade 7

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the divisors of $51$.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Using the bounds $6 < x < 22$ to pick the age.

💡Takeaway. Peel off the remainder and the divisor shows up as a factor -- then the bounds pick the right one.

Variant 11 hard

Maya's father is $58$ years old. When her father's age is divided by Maya's age, the remainder is $6$ . Maya's older sister, Julia, is $18$ years old. How old is Maya? (Ages are whole numbers of years.)

Your answer

Show solution

1 · Understandwhat's really being asked

Dividing $58$ by Maya's age leaves a remainder of $6$ , and Maya is younger than her $18$ -year-old sister. Find Maya's age.

Givens

Maya's father is $58$ years old.
$58$ divided by Maya's age leaves a remainder of $6$ .
Maya's older sister Julia is $18$ years old.

Unknowns

Maya's age.

Constraints

A remainder is always smaller than the divisor, so Maya is older than $6$ .
Maya is younger than her older sister, so she is younger than $18$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #2 Make a Systematic List#3 Eliminate Possibilities

Undo the division: taking the remainder away leaves an exact multiple of Maya's age. List the divisors of that number, then keep only the one that fits between the two bounds.

3 · Execute3 carry out the plan

1Take the remainder away

#11 Work Backwards 4.OA.B.4

If Maya is

x

years old and the quotient is

Q

, then

58 = x \times Q + 6

. Removing the

6

left over leaves a number that

x

divides exactly.

58 - 6 = 52 = x \times Q

Maya's age must be a divisor of

52

.

2List the divisors of 52

#2 Make a Systematic List 4.OA.B.4

Write

52

as a product in every way.

52 = 1 \times 52 = 2 \times 26 = 4 \times 13

The divisors are

1

,

2

,

4

,

13

,

26

,

52

.

3Apply both bounds

#3 Eliminate Possibilities 7.EE.B.4

The divisor must be bigger than the remainder

6

, and Maya is younger than

18

. Only one divisor sits in that range.

6 < x < 18 \Rightarrow x = 13

Maya is

13

.

Answer: 13 years old

4 · Reviewdoes it hold up?

Check: $13 \times 4 + 6 = 58$ , and $6 < 13$ , so $58$ divided by $13$ really leaves $6$ . Also $13 < 18$ .

Another way: Guess and check the ages $7$ to $17$ : dividing $58$ by each, only $13$ leaves a remainder of $6$ .

Standardsmin grade 7

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the divisors of $52$.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Using the bounds $6 < x < 18$ to pick the age.

💡Takeaway. Peel off the remainder and the divisor shows up as a factor -- then the bounds pick the right one.

Variant 12 hard

Elena's father is $60$ years old. When her father's age is divided by Elena's age, the remainder is $12$ . Elena's older sister, Sara, is $24$ years old. How old is Elena? (Ages are whole numbers of years.)

Your answer

Show solution

1 · Understandwhat's really being asked

Dividing $60$ by Elena's age leaves a remainder of $12$ , and Elena is younger than her $24$ -year-old sister. Find Elena's age.

Givens

Elena's father is $60$ years old.
$60$ divided by Elena's age leaves a remainder of $12$ .
Elena's older sister Sara is $24$ years old.

Unknowns

Elena's age.

Constraints

A remainder is always smaller than the divisor, so Elena is older than $12$ .
Elena is younger than her older sister, so she is younger than $24$ .

2 · Planchoose the strategy

#11 Work Backwards · also uses: #2 Make a Systematic List#3 Eliminate Possibilities

Undo the division: taking the remainder away leaves an exact multiple of Elena's age. List the divisors of that number, then keep only the one that fits between the two bounds.

3 · Execute3 carry out the plan

1Take the remainder away

#11 Work Backwards 4.OA.B.4

If Elena is

x

years old and the quotient is

Q

, then

60 = x \times Q + 12

. Removing the

12

left over leaves a number that

x

divides exactly.

60 - 12 = 48 = x \times Q

Elena's age must be a divisor of

48

.

2List the divisors of 48

#2 Make a Systematic List 4.OA.B.4

Write

48

as a product in every way.

48 = 1 \times 48 = 2 \times 24 = 3 \times 16 = 4 \times 12 = 6 \times 8

The divisors are

1

,

2

,

3

,

4

,

6

,

8

,

12

,

16

,

24

,

48

.

3Apply both bounds

#3 Eliminate Possibilities 7.EE.B.4

The divisor must be bigger than the remainder

12

, and Elena is younger than

24

. Only one divisor sits in that range.

12 < x < 24 \Rightarrow x = 16

Elena is

16

.

Answer: 16 years old

4 · Reviewdoes it hold up?

Check: $16 \times 3 + 12 = 60$ , and $12 < 16$ , so $60$ divided by $16$ really leaves $12$ . Also $16 < 24$ .

Another way: Guess and check the ages $13$ to $23$ : dividing $60$ by each, only $16$ leaves a remainder of $12$ .

Standardsmin grade 7

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Finding the divisors of $48$.
7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Using the bounds $12 < x < 24$ to pick the age.

💡Takeaway. Peel off the remainder and the divisor shows up as a factor -- then the bounds pick the right one.

Divisor of dividend minus remainder, within bounds · 12 practice problems

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