← Average times count gives the total · Average from Total

Average times count gives the total · 12 practice problems

6.SP.B.56.SP.A.34.OA.A.3

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: Emma 10, Sophia 18, Olivia 26

The average number of marbles that Emma and Sophia have together is $14$ , and the average number of marbles that Sophia and Olivia have together is $22$ . If the average number of marbles for all three of them is $18$ , find how many marbles each of the three has.

Show solution

1 · Understandwhat's really being asked

Emma and Sophia average 14 marbles, Sophia and Olivia average 22, and all three average 18. We need each person's count.

Givens

Emma and Sophia average 14.
Sophia and Olivia average 22.
All three average 18.

Unknowns

How many marbles Emma, Sophia and Olivia each have.

Constraints

Sophia appears in both pairs, which is what links them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #11 Work Backwards

An average times its count is a total, and totals subtract cleanly. Each pair total taken from the three-person total leaves exactly the person who was not in that pair.

3 · Execute4 carry out the plan

1Turn each average into a total

#7 Identify Subproblems 6.SP.B.5

Multiply each average by how many people it covers.

14 \times 2 = 28,\quad 22 \times 2 = 44,\quad 18 \times 3 = 54

Three totals instead of three averages.

2Find Olivia by removing the Emma+Sophia pair

#11 Work Backwards 4.OA.A.3

The three-person total minus Emma and Sophia's total is whatever Olivia has.

54 - 28 = 26

Olivia has 26.

3Find Emma by removing the Sophia+Olivia pair

#11 Work Backwards 4.OA.A.3

The same move with the other pair leaves Emma.

54 - 44 = 10

Emma has 10.

4Find Sophia from the Emma+Sophia total

#11 Work Backwards 4.OA.A.3

Emma and Sophia have 28 between them, and Emma's share is now known.

28 - 10 = 18

Sophia has 18.

Answer: Emma 10, Sophia 18, Olivia 26

4 · Reviewdoes it hold up?

Check all three statements: (10 + 18) / 2 = 14, (18 + 26) / 2 = 22, and (10 + 18 + 26) / 3 = 18. All hold.

Another way: Subtracting the two pair totals gives 28 - 44 = -16, which is how much more Emma has than Olivia -- another route into the same three numbers.

Standardsmin grade 6

6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Reading each average as a total over its own count.
6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Treating the average as a summary that multiplies back out.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Subtracting totals to isolate one person at a time.

💡Takeaway. Averages resist arithmetic; totals do not. Multiply each average back into a total and the subtractions do the work.

Variant 2 easy answer: Emma 12, Sophia 20, Olivia 28

The average number of marbles that Emma and Sophia have together is $16$ , and the average number of marbles that Sophia and Olivia have together is $24$ . If the average number of marbles for all three of them is $20$ , find how many marbles each of the three has.

Show solution

1 · Understandwhat's really being asked

Emma and Sophia average 16 marbles, Sophia and Olivia average 24, and all three average 20. We need each person's count.

Givens

Emma and Sophia average 16.
Sophia and Olivia average 24.
All three average 20.

Unknowns

How many marbles Emma, Sophia and Olivia each have.

Constraints

Sophia appears in both pairs, which is what links them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #11 Work Backwards

An average times its count is a total, and totals subtract cleanly. Each pair total taken from the three-person total leaves exactly the person who was not in that pair.

3 · Execute4 carry out the plan

1Turn each average into a total

#7 Identify Subproblems 6.SP.B.5

Multiply each average by how many people it covers.

16 \times 2 = 32,\quad 24 \times 2 = 48,\quad 20 \times 3 = 60

Three totals instead of three averages.

2Find Olivia by removing the Emma+Sophia pair

#11 Work Backwards 4.OA.A.3

The three-person total minus Emma and Sophia's total is whatever Olivia has.

60 - 32 = 28

Olivia has 28.

3Find Emma by removing the Sophia+Olivia pair

#11 Work Backwards 4.OA.A.3

The same move with the other pair leaves Emma.

60 - 48 = 12

Emma has 12.

4Find Sophia from the Emma+Sophia total

#11 Work Backwards 4.OA.A.3

Emma and Sophia have 32 between them, and Emma's share is now known.

32 - 12 = 20

Sophia has 20.

Answer: Emma 12, Sophia 20, Olivia 28

4 · Reviewdoes it hold up?

Check all three statements: (12 + 20) / 2 = 16, (20 + 28) / 2 = 24, and (12 + 20 + 28) / 3 = 20. All hold.

Another way: Subtracting the two pair totals gives 32 - 48 = -16, which is how much more Emma has than Olivia -- another route into the same three numbers.

Standardsmin grade 6

6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Reading each average as a total over its own count.
6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Treating the average as a summary that multiplies back out.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Subtracting totals to isolate one person at a time.

💡Takeaway. Averages resist arithmetic; totals do not. Multiply each average back into a total and the subtractions do the work.

Variant 3 easy answer: Emma 14, Sophia 22, Olivia 30

The average number of marbles that Emma and Sophia have together is $18$ , and the average number of marbles that Sophia and Olivia have together is $26$ . If the average number of marbles for all three of them is $22$ , find how many marbles each of the three has.

Show solution

1 · Understandwhat's really being asked

Emma and Sophia average 18 marbles, Sophia and Olivia average 26, and all three average 22. We need each person's count.

Givens

Emma and Sophia average 18.
Sophia and Olivia average 26.
All three average 22.

Unknowns

How many marbles Emma, Sophia and Olivia each have.

Constraints

Sophia appears in both pairs, which is what links them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #11 Work Backwards

An average times its count is a total, and totals subtract cleanly. Each pair total taken from the three-person total leaves exactly the person who was not in that pair.

3 · Execute4 carry out the plan

1Turn each average into a total

#7 Identify Subproblems 6.SP.B.5

Multiply each average by how many people it covers.

18 \times 2 = 36,\quad 26 \times 2 = 52,\quad 22 \times 3 = 66

Three totals instead of three averages.

2Find Olivia by removing the Emma+Sophia pair

#11 Work Backwards 4.OA.A.3

The three-person total minus Emma and Sophia's total is whatever Olivia has.

66 - 36 = 30

Olivia has 30.

3Find Emma by removing the Sophia+Olivia pair

#11 Work Backwards 4.OA.A.3

The same move with the other pair leaves Emma.

66 - 52 = 14

Emma has 14.

4Find Sophia from the Emma+Sophia total

#11 Work Backwards 4.OA.A.3

Emma and Sophia have 36 between them, and Emma's share is now known.

36 - 14 = 22

Sophia has 22.

Answer: Emma 14, Sophia 22, Olivia 30

4 · Reviewdoes it hold up?

Check all three statements: (14 + 22) / 2 = 18, (22 + 30) / 2 = 26, and (14 + 22 + 30) / 3 = 22. All hold.

Another way: Subtracting the two pair totals gives 36 - 52 = -16, which is how much more Emma has than Olivia -- another route into the same three numbers.

Standardsmin grade 6

6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Reading each average as a total over its own count.
6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Treating the average as a summary that multiplies back out.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Subtracting totals to isolate one person at a time.

💡Takeaway. Averages resist arithmetic; totals do not. Multiply each average back into a total and the subtractions do the work.

Variant 4 easy answer: Emma 16, Sophia 24, Olivia 32

The average number of marbles that Emma and Sophia have together is $20$ , and the average number of marbles that Sophia and Olivia have together is $28$ . If the average number of marbles for all three of them is $24$ , find how many marbles each of the three has.

Show solution

1 · Understandwhat's really being asked

Emma and Sophia average 20 marbles, Sophia and Olivia average 28, and all three average 24. We need each person's count.

Givens

Emma and Sophia average 20.
Sophia and Olivia average 28.
All three average 24.

Unknowns

How many marbles Emma, Sophia and Olivia each have.

Constraints

Sophia appears in both pairs, which is what links them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #11 Work Backwards

An average times its count is a total, and totals subtract cleanly. Each pair total taken from the three-person total leaves exactly the person who was not in that pair.

3 · Execute4 carry out the plan

1Turn each average into a total

#7 Identify Subproblems 6.SP.B.5

Multiply each average by how many people it covers.

20 \times 2 = 40,\quad 28 \times 2 = 56,\quad 24 \times 3 = 72

Three totals instead of three averages.

2Find Olivia by removing the Emma+Sophia pair

#11 Work Backwards 4.OA.A.3

The three-person total minus Emma and Sophia's total is whatever Olivia has.

72 - 40 = 32

Olivia has 32.

3Find Emma by removing the Sophia+Olivia pair

#11 Work Backwards 4.OA.A.3

The same move with the other pair leaves Emma.

72 - 56 = 16

Emma has 16.

4Find Sophia from the Emma+Sophia total

#11 Work Backwards 4.OA.A.3

Emma and Sophia have 40 between them, and Emma's share is now known.

40 - 16 = 24

Sophia has 24.

Answer: Emma 16, Sophia 24, Olivia 32

4 · Reviewdoes it hold up?

Check all three statements: (16 + 24) / 2 = 20, (24 + 32) / 2 = 28, and (16 + 24 + 32) / 3 = 24. All hold.

Another way: Subtracting the two pair totals gives 40 - 56 = -16, which is how much more Emma has than Olivia -- another route into the same three numbers.

Standardsmin grade 6

6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Reading each average as a total over its own count.
6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Treating the average as a summary that multiplies back out.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Subtracting totals to isolate one person at a time.

💡Takeaway. Averages resist arithmetic; totals do not. Multiply each average back into a total and the subtractions do the work.

Variant 5 medium answer: Emma 20, Sophia 26, Olivia 32

The average number of marbles that Emma and Sophia have together is $23$ , and the average number of marbles that Sophia and Olivia have together is $29$ . If the average number of marbles for all three of them is $26$ , find how many marbles each of the three has.

Show solution

1 · Understandwhat's really being asked

Emma and Sophia average 23 marbles, Sophia and Olivia average 29, and all three average 26. We need each person's count.

Givens

Emma and Sophia average 23.
Sophia and Olivia average 29.
All three average 26.

Unknowns

How many marbles Emma, Sophia and Olivia each have.

Constraints

Sophia appears in both pairs, which is what links them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #11 Work Backwards

An average times its count is a total, and totals subtract cleanly. Each pair total taken from the three-person total leaves exactly the person who was not in that pair.

3 · Execute4 carry out the plan

1Turn each average into a total

#7 Identify Subproblems 6.SP.B.5

Multiply each average by how many people it covers.

23 \times 2 = 46,\quad 29 \times 2 = 58,\quad 26 \times 3 = 78

Three totals instead of three averages.

2Find Olivia by removing the Emma+Sophia pair

#11 Work Backwards 4.OA.A.3

The three-person total minus Emma and Sophia's total is whatever Olivia has.

78 - 46 = 32

Olivia has 32.

3Find Emma by removing the Sophia+Olivia pair

#11 Work Backwards 4.OA.A.3

The same move with the other pair leaves Emma.

78 - 58 = 20

Emma has 20.

4Find Sophia from the Emma+Sophia total

#11 Work Backwards 4.OA.A.3

Emma and Sophia have 46 between them, and Emma's share is now known.

46 - 20 = 26

Sophia has 26.

Answer: Emma 20, Sophia 26, Olivia 32

4 · Reviewdoes it hold up?

Check all three statements: (20 + 26) / 2 = 23, (26 + 32) / 2 = 29, and (20 + 26 + 32) / 3 = 26. All hold.

Another way: Subtracting the two pair totals gives 46 - 58 = -12, which is how much more Emma has than Olivia -- another route into the same three numbers.

Standardsmin grade 6

6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Reading each average as a total over its own count.
6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Treating the average as a summary that multiplies back out.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Subtracting totals to isolate one person at a time.

💡Takeaway. Averages resist arithmetic; totals do not. Multiply each average back into a total and the subtractions do the work.

Variant 6 medium answer: Emma 22, Sophia 32, Olivia 42

The average number of marbles that Emma and Sophia have together is $27$ , and the average number of marbles that Sophia and Olivia have together is $37$ . If the average number of marbles for all three of them is $32$ , find how many marbles each of the three has.

Show solution

1 · Understandwhat's really being asked

Emma and Sophia average 27 marbles, Sophia and Olivia average 37, and all three average 32. We need each person's count.

Givens

Emma and Sophia average 27.
Sophia and Olivia average 37.
All three average 32.

Unknowns

How many marbles Emma, Sophia and Olivia each have.

Constraints

Sophia appears in both pairs, which is what links them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #11 Work Backwards

An average times its count is a total, and totals subtract cleanly. Each pair total taken from the three-person total leaves exactly the person who was not in that pair.

3 · Execute4 carry out the plan

1Turn each average into a total

#7 Identify Subproblems 6.SP.B.5

Multiply each average by how many people it covers.

27 \times 2 = 54,\quad 37 \times 2 = 74,\quad 32 \times 3 = 96

Three totals instead of three averages.

2Find Olivia by removing the Emma+Sophia pair

#11 Work Backwards 4.OA.A.3

The three-person total minus Emma and Sophia's total is whatever Olivia has.

96 - 54 = 42

Olivia has 42.

3Find Emma by removing the Sophia+Olivia pair

#11 Work Backwards 4.OA.A.3

The same move with the other pair leaves Emma.

96 - 74 = 22

Emma has 22.

4Find Sophia from the Emma+Sophia total

#11 Work Backwards 4.OA.A.3

Emma and Sophia have 54 between them, and Emma's share is now known.

54 - 22 = 32

Sophia has 32.

Answer: Emma 22, Sophia 32, Olivia 42

4 · Reviewdoes it hold up?

Check all three statements: (22 + 32) / 2 = 27, (32 + 42) / 2 = 37, and (22 + 32 + 42) / 3 = 32. All hold.

Another way: Subtracting the two pair totals gives 54 - 74 = -20, which is how much more Emma has than Olivia -- another route into the same three numbers.

Standardsmin grade 6

6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Reading each average as a total over its own count.
6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Treating the average as a summary that multiplies back out.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Subtracting totals to isolate one person at a time.

💡Takeaway. Averages resist arithmetic; totals do not. Multiply each average back into a total and the subtractions do the work.

Variant 7 medium answer: Emma 36, Sophia 40, Olivia 38

The average number of marbles that Emma and Sophia have together is $38$ , and the average number of marbles that Sophia and Olivia have together is $39$ . If the average number of marbles for all three of them is $38$ , find how many marbles each of the three has.

Show solution

1 · Understandwhat's really being asked

Emma and Sophia average 38 marbles, Sophia and Olivia average 39, and all three average 38. We need each person's count.

Givens

Emma and Sophia average 38.
Sophia and Olivia average 39.
All three average 38.

Unknowns

How many marbles Emma, Sophia and Olivia each have.

Constraints

Sophia appears in both pairs, which is what links them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #11 Work Backwards

An average times its count is a total, and totals subtract cleanly. Each pair total taken from the three-person total leaves exactly the person who was not in that pair.

3 · Execute4 carry out the plan

1Turn each average into a total

#7 Identify Subproblems 6.SP.B.5

Multiply each average by how many people it covers.

38 \times 2 = 76,\quad 39 \times 2 = 78,\quad 38 \times 3 = 114

Three totals instead of three averages.

2Find Olivia by removing the Emma+Sophia pair

#11 Work Backwards 4.OA.A.3

The three-person total minus Emma and Sophia's total is whatever Olivia has.

114 - 76 = 38

Olivia has 38.

3Find Emma by removing the Sophia+Olivia pair

#11 Work Backwards 4.OA.A.3

The same move with the other pair leaves Emma.

114 - 78 = 36

Emma has 36.

4Find Sophia from the Emma+Sophia total

#11 Work Backwards 4.OA.A.3

Emma and Sophia have 76 between them, and Emma's share is now known.

76 - 36 = 40

Sophia has 40.

Answer: Emma 36, Sophia 40, Olivia 38

4 · Reviewdoes it hold up?

Check all three statements: (36 + 40) / 2 = 38, (40 + 38) / 2 = 39, and (36 + 40 + 38) / 3 = 38. All hold.

Another way: Subtracting the two pair totals gives 76 - 78 = -2, which is how much more Emma has than Olivia -- another route into the same three numbers.

Standardsmin grade 6

6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Reading each average as a total over its own count.
6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Treating the average as a summary that multiplies back out.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Subtracting totals to isolate one person at a time.

💡Takeaway. Averages resist arithmetic; totals do not. Multiply each average back into a total and the subtractions do the work.

Variant 8 medium answer: Emma 28, Sophia 34, Olivia 46

The average number of marbles that Emma and Sophia have together is $31$ , and the average number of marbles that Sophia and Olivia have together is $40$ . If the average number of marbles for all three of them is $36$ , find how many marbles each of the three has.

Show solution

1 · Understandwhat's really being asked

Emma and Sophia average 31 marbles, Sophia and Olivia average 40, and all three average 36. We need each person's count.

Givens

Emma and Sophia average 31.
Sophia and Olivia average 40.
All three average 36.

Unknowns

How many marbles Emma, Sophia and Olivia each have.

Constraints

Sophia appears in both pairs, which is what links them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #11 Work Backwards

An average times its count is a total, and totals subtract cleanly. Each pair total taken from the three-person total leaves exactly the person who was not in that pair.

3 · Execute4 carry out the plan

1Turn each average into a total

#7 Identify Subproblems 6.SP.B.5

Multiply each average by how many people it covers.

31 \times 2 = 62,\quad 40 \times 2 = 80,\quad 36 \times 3 = 108

Three totals instead of three averages.

2Find Olivia by removing the Emma+Sophia pair

#11 Work Backwards 4.OA.A.3

The three-person total minus Emma and Sophia's total is whatever Olivia has.

108 - 62 = 46

Olivia has 46.

3Find Emma by removing the Sophia+Olivia pair

#11 Work Backwards 4.OA.A.3

The same move with the other pair leaves Emma.

108 - 80 = 28

Emma has 28.

4Find Sophia from the Emma+Sophia total

#11 Work Backwards 4.OA.A.3

Emma and Sophia have 62 between them, and Emma's share is now known.

62 - 28 = 34

Sophia has 34.

Answer: Emma 28, Sophia 34, Olivia 46

4 · Reviewdoes it hold up?

Check all three statements: (28 + 34) / 2 = 31, (34 + 46) / 2 = 40, and (28 + 34 + 46) / 3 = 36. All hold.

Another way: Subtracting the two pair totals gives 62 - 80 = -18, which is how much more Emma has than Olivia -- another route into the same three numbers.

Standardsmin grade 6

6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Reading each average as a total over its own count.
6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Treating the average as a summary that multiplies back out.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Subtracting totals to isolate one person at a time.

💡Takeaway. Averages resist arithmetic; totals do not. Multiply each average back into a total and the subtractions do the work.

Variant 9 hard answer: Emma 42, Sophia 48, Olivia 54

The average number of marbles that Emma and Sophia have together is $45$ , and the average number of marbles that Sophia and Olivia have together is $51$ . If the average number of marbles for all three of them is $48$ , find how many marbles each of the three has.

Show solution

1 · Understandwhat's really being asked

Emma and Sophia average 45 marbles, Sophia and Olivia average 51, and all three average 48. We need each person's count.

Givens

Emma and Sophia average 45.
Sophia and Olivia average 51.
All three average 48.

Unknowns

How many marbles Emma, Sophia and Olivia each have.

Constraints

Sophia appears in both pairs, which is what links them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #11 Work Backwards

An average times its count is a total, and totals subtract cleanly. Each pair total taken from the three-person total leaves exactly the person who was not in that pair.

3 · Execute4 carry out the plan

1Turn each average into a total

#7 Identify Subproblems 6.SP.B.5

Multiply each average by how many people it covers.

45 \times 2 = 90,\quad 51 \times 2 = 102,\quad 48 \times 3 = 144

Three totals instead of three averages.

2Find Olivia by removing the Emma+Sophia pair

#11 Work Backwards 4.OA.A.3

The three-person total minus Emma and Sophia's total is whatever Olivia has.

144 - 90 = 54

Olivia has 54.

3Find Emma by removing the Sophia+Olivia pair

#11 Work Backwards 4.OA.A.3

The same move with the other pair leaves Emma.

144 - 102 = 42

Emma has 42.

4Find Sophia from the Emma+Sophia total

#11 Work Backwards 4.OA.A.3

Emma and Sophia have 90 between them, and Emma's share is now known.

90 - 42 = 48

Sophia has 48.

Answer: Emma 42, Sophia 48, Olivia 54

4 · Reviewdoes it hold up?

Check all three statements: (42 + 48) / 2 = 45, (48 + 54) / 2 = 51, and (42 + 48 + 54) / 3 = 48. All hold.

Another way: Subtracting the two pair totals gives 90 - 102 = -12, which is how much more Emma has than Olivia -- another route into the same three numbers.

Standardsmin grade 6

6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Reading each average as a total over its own count.
6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Treating the average as a summary that multiplies back out.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Subtracting totals to isolate one person at a time.

💡Takeaway. Averages resist arithmetic; totals do not. Multiply each average back into a total and the subtractions do the work.

Variant 10 hard answer: Emma 44, Sophia 50, Olivia 62

The average number of marbles that Emma and Sophia have together is $47$ , and the average number of marbles that Sophia and Olivia have together is $56$ . If the average number of marbles for all three of them is $52$ , find how many marbles each of the three has.

Show solution

1 · Understandwhat's really being asked

Emma and Sophia average 47 marbles, Sophia and Olivia average 56, and all three average 52. We need each person's count.

Givens

Emma and Sophia average 47.
Sophia and Olivia average 56.
All three average 52.

Unknowns

How many marbles Emma, Sophia and Olivia each have.

Constraints

Sophia appears in both pairs, which is what links them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #11 Work Backwards

An average times its count is a total, and totals subtract cleanly. Each pair total taken from the three-person total leaves exactly the person who was not in that pair.

3 · Execute4 carry out the plan

1Turn each average into a total

#7 Identify Subproblems 6.SP.B.5

Multiply each average by how many people it covers.

47 \times 2 = 94,\quad 56 \times 2 = 112,\quad 52 \times 3 = 156

Three totals instead of three averages.

2Find Olivia by removing the Emma+Sophia pair

#11 Work Backwards 4.OA.A.3

The three-person total minus Emma and Sophia's total is whatever Olivia has.

156 - 94 = 62

Olivia has 62.

3Find Emma by removing the Sophia+Olivia pair

#11 Work Backwards 4.OA.A.3

The same move with the other pair leaves Emma.

156 - 112 = 44

Emma has 44.

4Find Sophia from the Emma+Sophia total

#11 Work Backwards 4.OA.A.3

Emma and Sophia have 94 between them, and Emma's share is now known.

94 - 44 = 50

Sophia has 50.

Answer: Emma 44, Sophia 50, Olivia 62

4 · Reviewdoes it hold up?

Check all three statements: (44 + 50) / 2 = 47, (50 + 62) / 2 = 56, and (44 + 50 + 62) / 3 = 52. All hold.

Another way: Subtracting the two pair totals gives 94 - 112 = -18, which is how much more Emma has than Olivia -- another route into the same three numbers.

Standardsmin grade 6

6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Reading each average as a total over its own count.
6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Treating the average as a summary that multiplies back out.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Subtracting totals to isolate one person at a time.

💡Takeaway. Averages resist arithmetic; totals do not. Multiply each average back into a total and the subtractions do the work.

Variant 11 hard answer: Emma 52, Sophia 56, Olivia 60

The average number of marbles that Emma and Sophia have together is $54$ , and the average number of marbles that Sophia and Olivia have together is $58$ . If the average number of marbles for all three of them is $56$ , find how many marbles each of the three has.

Show solution

1 · Understandwhat's really being asked

Emma and Sophia average 54 marbles, Sophia and Olivia average 58, and all three average 56. We need each person's count.

Givens

Emma and Sophia average 54.
Sophia and Olivia average 58.
All three average 56.

Unknowns

How many marbles Emma, Sophia and Olivia each have.

Constraints

Sophia appears in both pairs, which is what links them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #11 Work Backwards

An average times its count is a total, and totals subtract cleanly. Each pair total taken from the three-person total leaves exactly the person who was not in that pair.

3 · Execute4 carry out the plan

1Turn each average into a total

#7 Identify Subproblems 6.SP.B.5

Multiply each average by how many people it covers.

54 \times 2 = 108,\quad 58 \times 2 = 116,\quad 56 \times 3 = 168

Three totals instead of three averages.

2Find Olivia by removing the Emma+Sophia pair

#11 Work Backwards 4.OA.A.3

The three-person total minus Emma and Sophia's total is whatever Olivia has.

168 - 108 = 60

Olivia has 60.

3Find Emma by removing the Sophia+Olivia pair

#11 Work Backwards 4.OA.A.3

The same move with the other pair leaves Emma.

168 - 116 = 52

Emma has 52.

4Find Sophia from the Emma+Sophia total

#11 Work Backwards 4.OA.A.3

Emma and Sophia have 108 between them, and Emma's share is now known.

108 - 52 = 56

Sophia has 56.

Answer: Emma 52, Sophia 56, Olivia 60

4 · Reviewdoes it hold up?

Check all three statements: (52 + 56) / 2 = 54, (56 + 60) / 2 = 58, and (52 + 56 + 60) / 3 = 56. All hold.

Another way: Subtracting the two pair totals gives 108 - 116 = -8, which is how much more Emma has than Olivia -- another route into the same three numbers.

Standardsmin grade 6

6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Reading each average as a total over its own count.
6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Treating the average as a summary that multiplies back out.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Subtracting totals to isolate one person at a time.

💡Takeaway. Averages resist arithmetic; totals do not. Multiply each average back into a total and the subtractions do the work.

Variant 12 hard answer: Emma 58, Sophia 64, Olivia 70

The average number of marbles that Emma and Sophia have together is $61$ , and the average number of marbles that Sophia and Olivia have together is $67$ . If the average number of marbles for all three of them is $64$ , find how many marbles each of the three has.

Show solution

1 · Understandwhat's really being asked

Emma and Sophia average 61 marbles, Sophia and Olivia average 67, and all three average 64. We need each person's count.

Givens

Emma and Sophia average 61.
Sophia and Olivia average 67.
All three average 64.

Unknowns

How many marbles Emma, Sophia and Olivia each have.

Constraints

Sophia appears in both pairs, which is what links them.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #11 Work Backwards

An average times its count is a total, and totals subtract cleanly. Each pair total taken from the three-person total leaves exactly the person who was not in that pair.

3 · Execute4 carry out the plan

1Turn each average into a total

#7 Identify Subproblems 6.SP.B.5

Multiply each average by how many people it covers.

61 \times 2 = 122,\quad 67 \times 2 = 134,\quad 64 \times 3 = 192

Three totals instead of three averages.

2Find Olivia by removing the Emma+Sophia pair

#11 Work Backwards 4.OA.A.3

The three-person total minus Emma and Sophia's total is whatever Olivia has.

192 - 122 = 70

Olivia has 70.

3Find Emma by removing the Sophia+Olivia pair

#11 Work Backwards 4.OA.A.3

The same move with the other pair leaves Emma.

192 - 134 = 58

Emma has 58.

4Find Sophia from the Emma+Sophia total

#11 Work Backwards 4.OA.A.3

Emma and Sophia have 122 between them, and Emma's share is now known.

122 - 58 = 64

Sophia has 64.

Answer: Emma 58, Sophia 64, Olivia 70

4 · Reviewdoes it hold up?

Check all three statements: (58 + 64) / 2 = 61, (64 + 70) / 2 = 67, and (58 + 64 + 70) / 3 = 64. All hold.

Another way: Subtracting the two pair totals gives 122 - 134 = -12, which is how much more Emma has than Olivia -- another route into the same three numbers.

Standardsmin grade 6

6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Reading each average as a total over its own count.
6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Treating the average as a summary that multiplies back out.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Subtracting totals to isolate one person at a time.

💡Takeaway. Averages resist arithmetic; totals do not. Multiply each average back into a total and the subtractions do the work.