← Set up one-unknown equation from mean · Average from Total

Set up one-unknown equation from mean · 12 practice problems

6.SP.A.36.SP.B.56.EE.B.7

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Generated variants — 12

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

Variant 1 easy

The table shows the number of cell phones a store sold each quarter. The average number of cell phones sold over the four quarters is $180$ , and the number sold in the 2nd quarter is $20$ fewer than the number sold in the 4th quarter. Find the number of cell phones sold in the 4th quarter.

Quarter	Q1	Q2	Q3	Q4
Phones sold	200		160

Your answer

Show solution

1 · Understandwhat's really being asked

Four quarters average 180 phones. Two quarters are on the table (200 and 160); the other two are blank, and the 2nd is 20 fewer than the 4th. We need the 4th quarter's number.

Givens

The average over the four quarters is 180.
Q1 sold 200 and Q3 sold 160.
Q2 is 20 fewer than Q4.

Unknowns

How many phones were sold in Q4.

Constraints

The average counts all four quarters, including the two unknown ones.

2 · Planchoose the strategy

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

An average is a total in disguise. Undo it to get the four-quarter total, subtract the two known quarters to learn what the blanks add to, then use the stated gap to split that sum.

3 · Execute3 carry out the plan

1Recover the total from the average

#11 Work Backwards 6.SP.A.3

An average of 180 over four quarters means the four numbers add to four lots of 180.

180 \times 4 = 720

The average is the total shared out evenly, so multiplying undoes the sharing.

2Find what Q2 and Q4 together must be

#7 Identify Subproblems 6.SP.B.5

Take the two quarters we know away from the total.

720 - 200 - 160 = 360

The two blanks add to 360.

3Share the 360 using the 20 gap

#7 Identify Subproblems 6.EE.B.7

If Q2 were as big as Q4, the pair would total 20 more, that is 380, split evenly between them.

(360 + 20) \div 2 = 190

Q4 is 190, so Q2 is 170.

Answer: 190 phones

4 · Reviewdoes it hold up?

Check both conditions: 200 + 170 + 160 + 190 = 720, and 720 divided by 4 is 180. Also 190 - 170 = 20.

Another way: Call Q4 the unknown: Q2 is that minus 20, so twice the unknown minus 20 equals 360, giving 190 again.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading the average of 180 as a total of 720 across four quarters.
6.SP.B.5 Summarize numerical data sets in relation to their context — Relating the table's known entries to the summary figure.
6.EE.B.7 Solve real-world problems by writing and solving equations of the form x + p = q — Splitting 360 into two parts that differ by 20.

💡Takeaway. An average is a total in disguise: multiply it back out, and a blank in a table becomes an ordinary missing-number problem.

Variant 2 easy

The table shows the number of cell phones a store sold each quarter. The average number of cell phones sold over the four quarters is $215$ , and the number sold in the 2nd quarter is $30$ fewer than the number sold in the 4th quarter. Find the number of cell phones sold in the 4th quarter.

Quarter	Q1	Q2	Q3	Q4
Phones sold	190		230

Your answer

Show solution

1 · Understandwhat's really being asked

Four quarters average 215 phones. Two quarters are on the table (190 and 230); the other two are blank, and the 2nd is 30 fewer than the 4th. We need the 4th quarter's number.

Givens

The average over the four quarters is 215.
Q1 sold 190 and Q3 sold 230.
Q2 is 30 fewer than Q4.

Unknowns

How many phones were sold in Q4.

Constraints

The average counts all four quarters, including the two unknown ones.

2 · Planchoose the strategy

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

An average is a total in disguise. Undo it to get the four-quarter total, subtract the two known quarters to learn what the blanks add to, then use the stated gap to split that sum.

3 · Execute3 carry out the plan

1Recover the total from the average

#11 Work Backwards 6.SP.A.3

An average of 215 over four quarters means the four numbers add to four lots of 215.

215 \times 4 = 860

The average is the total shared out evenly, so multiplying undoes the sharing.

2Find what Q2 and Q4 together must be

#7 Identify Subproblems 6.SP.B.5

Take the two quarters we know away from the total.

860 - 190 - 230 = 440

The two blanks add to 440.

3Share the 440 using the 30 gap

#7 Identify Subproblems 6.EE.B.7

If Q2 were as big as Q4, the pair would total 30 more, that is 470, split evenly between them.

(440 + 30) \div 2 = 235

Q4 is 235, so Q2 is 205.

Answer: 235 phones

4 · Reviewdoes it hold up?

Check both conditions: 190 + 205 + 230 + 235 = 860, and 860 divided by 4 is 215. Also 235 - 205 = 30.

Another way: Call Q4 the unknown: Q2 is that minus 30, so twice the unknown minus 30 equals 440, giving 235 again.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading the average of 215 as a total of 860 across four quarters.
6.SP.B.5 Summarize numerical data sets in relation to their context — Relating the table's known entries to the summary figure.
6.EE.B.7 Solve real-world problems by writing and solving equations of the form x + p = q — Splitting 440 into two parts that differ by 30.

💡Takeaway. An average is a total in disguise: multiply it back out, and a blank in a table becomes an ordinary missing-number problem.

Variant 3 easy

The table shows the number of cell phones a store sold each quarter. The average number of cell phones sold over the four quarters is $260$ , and the number sold in the 2nd quarter is $40$ fewer than the number sold in the 4th quarter. Find the number of cell phones sold in the 4th quarter.

Quarter	Q1	Q2	Q3	Q4
Phones sold	240		280

Your answer

Show solution

1 · Understandwhat's really being asked

Four quarters average 260 phones. Two quarters are on the table (240 and 280); the other two are blank, and the 2nd is 40 fewer than the 4th. We need the 4th quarter's number.

Givens

The average over the four quarters is 260.
Q1 sold 240 and Q3 sold 280.
Q2 is 40 fewer than Q4.

Unknowns

How many phones were sold in Q4.

Constraints

The average counts all four quarters, including the two unknown ones.

2 · Planchoose the strategy

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

An average is a total in disguise. Undo it to get the four-quarter total, subtract the two known quarters to learn what the blanks add to, then use the stated gap to split that sum.

3 · Execute3 carry out the plan

1Recover the total from the average

#11 Work Backwards 6.SP.A.3

An average of 260 over four quarters means the four numbers add to four lots of 260.

260 \times 4 = 1040

The average is the total shared out evenly, so multiplying undoes the sharing.

2Find what Q2 and Q4 together must be

#7 Identify Subproblems 6.SP.B.5

Take the two quarters we know away from the total.

1040 - 240 - 280 = 520

The two blanks add to 520.

3Share the 520 using the 40 gap

#7 Identify Subproblems 6.EE.B.7

If Q2 were as big as Q4, the pair would total 40 more, that is 560, split evenly between them.

(520 + 40) \div 2 = 280

Q4 is 280, so Q2 is 240.

Answer: 280 phones

4 · Reviewdoes it hold up?

Check both conditions: 240 + 240 + 280 + 280 = 1040, and 1040 divided by 4 is 260. Also 280 - 240 = 40.

Another way: Call Q4 the unknown: Q2 is that minus 40, so twice the unknown minus 40 equals 520, giving 280 again.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading the average of 260 as a total of 1040 across four quarters.
6.SP.B.5 Summarize numerical data sets in relation to their context — Relating the table's known entries to the summary figure.
6.EE.B.7 Solve real-world problems by writing and solving equations of the form x + p = q — Splitting 520 into two parts that differ by 40.

💡Takeaway. An average is a total in disguise: multiply it back out, and a blank in a table becomes an ordinary missing-number problem.

Variant 4 easy

The table shows the number of cell phones a store sold each quarter. The average number of cell phones sold over the four quarters is $275$ , and the number sold in the 2nd quarter is $50$ fewer than the number sold in the 4th quarter. Find the number of cell phones sold in the 4th quarter.

Quarter	Q1	Q2	Q3	Q4
Phones sold	250		300

Your answer

Show solution

1 · Understandwhat's really being asked

Four quarters average 275 phones. Two quarters are on the table (250 and 300); the other two are blank, and the 2nd is 50 fewer than the 4th. We need the 4th quarter's number.

Givens

The average over the four quarters is 275.
Q1 sold 250 and Q3 sold 300.
Q2 is 50 fewer than Q4.

Unknowns

How many phones were sold in Q4.

Constraints

The average counts all four quarters, including the two unknown ones.

2 · Planchoose the strategy

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

An average is a total in disguise. Undo it to get the four-quarter total, subtract the two known quarters to learn what the blanks add to, then use the stated gap to split that sum.

3 · Execute3 carry out the plan

1Recover the total from the average

#11 Work Backwards 6.SP.A.3

An average of 275 over four quarters means the four numbers add to four lots of 275.

275 \times 4 = 1100

The average is the total shared out evenly, so multiplying undoes the sharing.

2Find what Q2 and Q4 together must be

#7 Identify Subproblems 6.SP.B.5

Take the two quarters we know away from the total.

1100 - 250 - 300 = 550

The two blanks add to 550.

3Share the 550 using the 50 gap

#7 Identify Subproblems 6.EE.B.7

If Q2 were as big as Q4, the pair would total 50 more, that is 600, split evenly between them.

(550 + 50) \div 2 = 300

Q4 is 300, so Q2 is 250.

Answer: 300 phones

4 · Reviewdoes it hold up?

Check both conditions: 250 + 250 + 300 + 300 = 1100, and 1100 divided by 4 is 275. Also 300 - 250 = 50.

Another way: Call Q4 the unknown: Q2 is that minus 50, so twice the unknown minus 50 equals 550, giving 300 again.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading the average of 275 as a total of 1100 across four quarters.
6.SP.B.5 Summarize numerical data sets in relation to their context — Relating the table's known entries to the summary figure.
6.EE.B.7 Solve real-world problems by writing and solving equations of the form x + p = q — Splitting 550 into two parts that differ by 50.

💡Takeaway. An average is a total in disguise: multiply it back out, and a blank in a table becomes an ordinary missing-number problem.

Variant 5 medium

The table shows the number of cell phones a store sold each quarter. The average number of cell phones sold over the four quarters is $320$ , and the number sold in the 2nd quarter is $60$ fewer than the number sold in the 4th quarter. Find the number of cell phones sold in the 4th quarter.

Quarter	Q1	Q2	Q3	Q4
Phones sold	300		350

Your answer

Show solution

1 · Understandwhat's really being asked

Four quarters average 320 phones. Two quarters are on the table (300 and 350); the other two are blank, and the 2nd is 60 fewer than the 4th. We need the 4th quarter's number.

Givens

The average over the four quarters is 320.
Q1 sold 300 and Q3 sold 350.
Q2 is 60 fewer than Q4.

Unknowns

How many phones were sold in Q4.

Constraints

The average counts all four quarters, including the two unknown ones.

2 · Planchoose the strategy

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

An average is a total in disguise. Undo it to get the four-quarter total, subtract the two known quarters to learn what the blanks add to, then use the stated gap to split that sum.

3 · Execute3 carry out the plan

1Recover the total from the average

#11 Work Backwards 6.SP.A.3

An average of 320 over four quarters means the four numbers add to four lots of 320.

320 \times 4 = 1280

The average is the total shared out evenly, so multiplying undoes the sharing.

2Find what Q2 and Q4 together must be

#7 Identify Subproblems 6.SP.B.5

Take the two quarters we know away from the total.

1280 - 300 - 350 = 630

The two blanks add to 630.

3Share the 630 using the 60 gap

#7 Identify Subproblems 6.EE.B.7

If Q2 were as big as Q4, the pair would total 60 more, that is 690, split evenly between them.

(630 + 60) \div 2 = 345

Q4 is 345, so Q2 is 285.

Answer: 345 phones

4 · Reviewdoes it hold up?

Check both conditions: 300 + 285 + 350 + 345 = 1280, and 1280 divided by 4 is 320. Also 345 - 285 = 60.

Another way: Call Q4 the unknown: Q2 is that minus 60, so twice the unknown minus 60 equals 630, giving 345 again.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading the average of 320 as a total of 1280 across four quarters.
6.SP.B.5 Summarize numerical data sets in relation to their context — Relating the table's known entries to the summary figure.
6.EE.B.7 Solve real-world problems by writing and solving equations of the form x + p = q — Splitting 630 into two parts that differ by 60.

💡Takeaway. An average is a total in disguise: multiply it back out, and a blank in a table becomes an ordinary missing-number problem.

Variant 6 medium

The table shows the number of cell phones a store sold each quarter. The average number of cell phones sold over the four quarters is $340$ , and the number sold in the 2nd quarter is $80$ fewer than the number sold in the 4th quarter. Find the number of cell phones sold in the 4th quarter.

Quarter	Q1	Q2	Q3	Q4
Phones sold	360		320

Your answer

Show solution

1 · Understandwhat's really being asked

Four quarters average 340 phones. Two quarters are on the table (360 and 320); the other two are blank, and the 2nd is 80 fewer than the 4th. We need the 4th quarter's number.

Givens

The average over the four quarters is 340.
Q1 sold 360 and Q3 sold 320.
Q2 is 80 fewer than Q4.

Unknowns

How many phones were sold in Q4.

Constraints

The average counts all four quarters, including the two unknown ones.

2 · Planchoose the strategy

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

An average is a total in disguise. Undo it to get the four-quarter total, subtract the two known quarters to learn what the blanks add to, then use the stated gap to split that sum.

3 · Execute3 carry out the plan

1Recover the total from the average

#11 Work Backwards 6.SP.A.3

An average of 340 over four quarters means the four numbers add to four lots of 340.

340 \times 4 = 1360

The average is the total shared out evenly, so multiplying undoes the sharing.

2Find what Q2 and Q4 together must be

#7 Identify Subproblems 6.SP.B.5

Take the two quarters we know away from the total.

1360 - 360 - 320 = 680

The two blanks add to 680.

3Share the 680 using the 80 gap

#7 Identify Subproblems 6.EE.B.7

If Q2 were as big as Q4, the pair would total 80 more, that is 760, split evenly between them.

(680 + 80) \div 2 = 380

Q4 is 380, so Q2 is 300.

Answer: 380 phones

4 · Reviewdoes it hold up?

Check both conditions: 360 + 300 + 320 + 380 = 1360, and 1360 divided by 4 is 340. Also 380 - 300 = 80.

Another way: Call Q4 the unknown: Q2 is that minus 80, so twice the unknown minus 80 equals 680, giving 380 again.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading the average of 340 as a total of 1360 across four quarters.
6.SP.B.5 Summarize numerical data sets in relation to their context — Relating the table's known entries to the summary figure.
6.EE.B.7 Solve real-world problems by writing and solving equations of the form x + p = q — Splitting 680 into two parts that differ by 80.

💡Takeaway. An average is a total in disguise: multiply it back out, and a blank in a table becomes an ordinary missing-number problem.

Variant 7 medium

The table shows the number of cell phones a store sold each quarter. The average number of cell phones sold over the four quarters is $400$ , and the number sold in the 2nd quarter is $60$ fewer than the number sold in the 4th quarter. Find the number of cell phones sold in the 4th quarter.

Quarter	Q1	Q2	Q3	Q4
Phones sold	380		420

Your answer

Show solution

1 · Understandwhat's really being asked

Four quarters average 400 phones. Two quarters are on the table (380 and 420); the other two are blank, and the 2nd is 60 fewer than the 4th. We need the 4th quarter's number.

Givens

The average over the four quarters is 400.
Q1 sold 380 and Q3 sold 420.
Q2 is 60 fewer than Q4.

Unknowns

How many phones were sold in Q4.

Constraints

The average counts all four quarters, including the two unknown ones.

2 · Planchoose the strategy

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

An average is a total in disguise. Undo it to get the four-quarter total, subtract the two known quarters to learn what the blanks add to, then use the stated gap to split that sum.

3 · Execute3 carry out the plan

1Recover the total from the average

#11 Work Backwards 6.SP.A.3

An average of 400 over four quarters means the four numbers add to four lots of 400.

400 \times 4 = 1600

The average is the total shared out evenly, so multiplying undoes the sharing.

2Find what Q2 and Q4 together must be

#7 Identify Subproblems 6.SP.B.5

Take the two quarters we know away from the total.

1600 - 380 - 420 = 800

The two blanks add to 800.

3Share the 800 using the 60 gap

#7 Identify Subproblems 6.EE.B.7

If Q2 were as big as Q4, the pair would total 60 more, that is 860, split evenly between them.

(800 + 60) \div 2 = 430

Q4 is 430, so Q2 is 370.

Answer: 430 phones

4 · Reviewdoes it hold up?

Check both conditions: 380 + 370 + 420 + 430 = 1600, and 1600 divided by 4 is 400. Also 430 - 370 = 60.

Another way: Call Q4 the unknown: Q2 is that minus 60, so twice the unknown minus 60 equals 800, giving 430 again.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading the average of 400 as a total of 1600 across four quarters.
6.SP.B.5 Summarize numerical data sets in relation to their context — Relating the table's known entries to the summary figure.
6.EE.B.7 Solve real-world problems by writing and solving equations of the form x + p = q — Splitting 800 into two parts that differ by 60.

💡Takeaway. An average is a total in disguise: multiply it back out, and a blank in a table becomes an ordinary missing-number problem.

Variant 8 medium

The table shows the number of cell phones a store sold each quarter. The average number of cell phones sold over the four quarters is $395$ , and the number sold in the 2nd quarter is $80$ fewer than the number sold in the 4th quarter. Find the number of cell phones sold in the 4th quarter.

Quarter	Q1	Q2	Q3	Q4
Phones sold	450		410

Your answer

Show solution

1 · Understandwhat's really being asked

Four quarters average 395 phones. Two quarters are on the table (450 and 410); the other two are blank, and the 2nd is 80 fewer than the 4th. We need the 4th quarter's number.

Givens

The average over the four quarters is 395.
Q1 sold 450 and Q3 sold 410.
Q2 is 80 fewer than Q4.

Unknowns

How many phones were sold in Q4.

Constraints

The average counts all four quarters, including the two unknown ones.

2 · Planchoose the strategy

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

An average is a total in disguise. Undo it to get the four-quarter total, subtract the two known quarters to learn what the blanks add to, then use the stated gap to split that sum.

3 · Execute3 carry out the plan

1Recover the total from the average

#11 Work Backwards 6.SP.A.3

An average of 395 over four quarters means the four numbers add to four lots of 395.

395 \times 4 = 1580

The average is the total shared out evenly, so multiplying undoes the sharing.

2Find what Q2 and Q4 together must be

#7 Identify Subproblems 6.SP.B.5

Take the two quarters we know away from the total.

1580 - 450 - 410 = 720

The two blanks add to 720.

3Share the 720 using the 80 gap

#7 Identify Subproblems 6.EE.B.7

If Q2 were as big as Q4, the pair would total 80 more, that is 800, split evenly between them.

(720 + 80) \div 2 = 400

Q4 is 400, so Q2 is 320.

Answer: 400 phones

4 · Reviewdoes it hold up?

Check both conditions: 450 + 320 + 410 + 400 = 1580, and 1580 divided by 4 is 395. Also 400 - 320 = 80.

Another way: Call Q4 the unknown: Q2 is that minus 80, so twice the unknown minus 80 equals 720, giving 400 again.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading the average of 395 as a total of 1580 across four quarters.
6.SP.B.5 Summarize numerical data sets in relation to their context — Relating the table's known entries to the summary figure.
6.EE.B.7 Solve real-world problems by writing and solving equations of the form x + p = q — Splitting 720 into two parts that differ by 80.

💡Takeaway. An average is a total in disguise: multiply it back out, and a blank in a table becomes an ordinary missing-number problem.

Variant 9 hard

The table shows the number of cell phones a store sold each quarter. The average number of cell phones sold over the four quarters is $450$ , and the number sold in the 2nd quarter is $120$ fewer than the number sold in the 4th quarter. Find the number of cell phones sold in the 4th quarter.

Quarter	Q1	Q2	Q3	Q4
Phones sold	400		500

Your answer

Show solution

1 · Understandwhat's really being asked

Four quarters average 450 phones. Two quarters are on the table (400 and 500); the other two are blank, and the 2nd is 120 fewer than the 4th. We need the 4th quarter's number.

Givens

The average over the four quarters is 450.
Q1 sold 400 and Q3 sold 500.
Q2 is 120 fewer than Q4.

Unknowns

How many phones were sold in Q4.

Constraints

The average counts all four quarters, including the two unknown ones.

2 · Planchoose the strategy

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

An average is a total in disguise. Undo it to get the four-quarter total, subtract the two known quarters to learn what the blanks add to, then use the stated gap to split that sum.

3 · Execute3 carry out the plan

1Recover the total from the average

#11 Work Backwards 6.SP.A.3

An average of 450 over four quarters means the four numbers add to four lots of 450.

450 \times 4 = 1800

The average is the total shared out evenly, so multiplying undoes the sharing.

2Find what Q2 and Q4 together must be

#7 Identify Subproblems 6.SP.B.5

Take the two quarters we know away from the total.

1800 - 400 - 500 = 900

The two blanks add to 900.

3Share the 900 using the 120 gap

#7 Identify Subproblems 6.EE.B.7

If Q2 were as big as Q4, the pair would total 120 more, that is 1020, split evenly between them.

(900 + 120) \div 2 = 510

Q4 is 510, so Q2 is 390.

Answer: 510 phones

4 · Reviewdoes it hold up?

Check both conditions: 400 + 390 + 500 + 510 = 1800, and 1800 divided by 4 is 450. Also 510 - 390 = 120.

Another way: Call Q4 the unknown: Q2 is that minus 120, so twice the unknown minus 120 equals 900, giving 510 again.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading the average of 450 as a total of 1800 across four quarters.
6.SP.B.5 Summarize numerical data sets in relation to their context — Relating the table's known entries to the summary figure.
6.EE.B.7 Solve real-world problems by writing and solving equations of the form x + p = q — Splitting 900 into two parts that differ by 120.

💡Takeaway. An average is a total in disguise: multiply it back out, and a blank in a table becomes an ordinary missing-number problem.

Variant 10 hard

The table shows the number of cell phones a store sold each quarter. The average number of cell phones sold over the four quarters is $500$ , and the number sold in the 2nd quarter is $100$ fewer than the number sold in the 4th quarter. Find the number of cell phones sold in the 4th quarter.

Quarter	Q1	Q2	Q3	Q4
Phones sold	520		480

Your answer

Show solution

1 · Understandwhat's really being asked

Four quarters average 500 phones. Two quarters are on the table (520 and 480); the other two are blank, and the 2nd is 100 fewer than the 4th. We need the 4th quarter's number.

Givens

The average over the four quarters is 500.
Q1 sold 520 and Q3 sold 480.
Q2 is 100 fewer than Q4.

Unknowns

How many phones were sold in Q4.

Constraints

The average counts all four quarters, including the two unknown ones.

2 · Planchoose the strategy

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

An average is a total in disguise. Undo it to get the four-quarter total, subtract the two known quarters to learn what the blanks add to, then use the stated gap to split that sum.

3 · Execute3 carry out the plan

1Recover the total from the average

#11 Work Backwards 6.SP.A.3

An average of 500 over four quarters means the four numbers add to four lots of 500.

500 \times 4 = 2000

The average is the total shared out evenly, so multiplying undoes the sharing.

2Find what Q2 and Q4 together must be

#7 Identify Subproblems 6.SP.B.5

Take the two quarters we know away from the total.

2000 - 520 - 480 = 1000

The two blanks add to 1000.

3Share the 1000 using the 100 gap

#7 Identify Subproblems 6.EE.B.7

If Q2 were as big as Q4, the pair would total 100 more, that is 1100, split evenly between them.

(1000 + 100) \div 2 = 550

Q4 is 550, so Q2 is 450.

Answer: 550 phones

4 · Reviewdoes it hold up?

Check both conditions: 520 + 450 + 480 + 550 = 2000, and 2000 divided by 4 is 500. Also 550 - 450 = 100.

Another way: Call Q4 the unknown: Q2 is that minus 100, so twice the unknown minus 100 equals 1000, giving 550 again.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading the average of 500 as a total of 2000 across four quarters.
6.SP.B.5 Summarize numerical data sets in relation to their context — Relating the table's known entries to the summary figure.
6.EE.B.7 Solve real-world problems by writing and solving equations of the form x + p = q — Splitting 1000 into two parts that differ by 100.

💡Takeaway. An average is a total in disguise: multiply it back out, and a blank in a table becomes an ordinary missing-number problem.

Variant 11 hard

The table shows the number of cell phones a store sold each quarter. The average number of cell phones sold over the four quarters is $520$ , and the number sold in the 2nd quarter is $160$ fewer than the number sold in the 4th quarter. Find the number of cell phones sold in the 4th quarter.

Quarter	Q1	Q2	Q3	Q4
Phones sold	500		540

Your answer

Show solution

1 · Understandwhat's really being asked

Four quarters average 520 phones. Two quarters are on the table (500 and 540); the other two are blank, and the 2nd is 160 fewer than the 4th. We need the 4th quarter's number.

Givens

The average over the four quarters is 520.
Q1 sold 500 and Q3 sold 540.
Q2 is 160 fewer than Q4.

Unknowns

How many phones were sold in Q4.

Constraints

The average counts all four quarters, including the two unknown ones.

2 · Planchoose the strategy

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

An average is a total in disguise. Undo it to get the four-quarter total, subtract the two known quarters to learn what the blanks add to, then use the stated gap to split that sum.

3 · Execute3 carry out the plan

1Recover the total from the average

#11 Work Backwards 6.SP.A.3

An average of 520 over four quarters means the four numbers add to four lots of 520.

520 \times 4 = 2080

The average is the total shared out evenly, so multiplying undoes the sharing.

2Find what Q2 and Q4 together must be

#7 Identify Subproblems 6.SP.B.5

Take the two quarters we know away from the total.

2080 - 500 - 540 = 1040

The two blanks add to 1040.

3Share the 1040 using the 160 gap

#7 Identify Subproblems 6.EE.B.7

If Q2 were as big as Q4, the pair would total 160 more, that is 1200, split evenly between them.

(1040 + 160) \div 2 = 600

Q4 is 600, so Q2 is 440.

Answer: 600 phones

4 · Reviewdoes it hold up?

Check both conditions: 500 + 440 + 540 + 600 = 2080, and 2080 divided by 4 is 520. Also 600 - 440 = 160.

Another way: Call Q4 the unknown: Q2 is that minus 160, so twice the unknown minus 160 equals 1040, giving 600 again.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading the average of 520 as a total of 2080 across four quarters.
6.SP.B.5 Summarize numerical data sets in relation to their context — Relating the table's known entries to the summary figure.
6.EE.B.7 Solve real-world problems by writing and solving equations of the form x + p = q — Splitting 1040 into two parts that differ by 160.

💡Takeaway. An average is a total in disguise: multiply it back out, and a blank in a table becomes an ordinary missing-number problem.

Variant 12 hard

The table shows the number of cell phones a store sold each quarter. The average number of cell phones sold over the four quarters is $610$ , and the number sold in the 2nd quarter is $140$ fewer than the number sold in the 4th quarter. Find the number of cell phones sold in the 4th quarter.

Quarter	Q1	Q2	Q3	Q4
Phones sold	580		640

Your answer

Show solution

1 · Understandwhat's really being asked

Four quarters average 610 phones. Two quarters are on the table (580 and 640); the other two are blank, and the 2nd is 140 fewer than the 4th. We need the 4th quarter's number.

Givens

The average over the four quarters is 610.
Q1 sold 580 and Q3 sold 640.
Q2 is 140 fewer than Q4.

Unknowns

How many phones were sold in Q4.

Constraints

The average counts all four quarters, including the two unknown ones.

2 · Planchoose the strategy

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

An average is a total in disguise. Undo it to get the four-quarter total, subtract the two known quarters to learn what the blanks add to, then use the stated gap to split that sum.

3 · Execute3 carry out the plan

1Recover the total from the average

#11 Work Backwards 6.SP.A.3

An average of 610 over four quarters means the four numbers add to four lots of 610.

610 \times 4 = 2440

The average is the total shared out evenly, so multiplying undoes the sharing.

2Find what Q2 and Q4 together must be

#7 Identify Subproblems 6.SP.B.5

Take the two quarters we know away from the total.

2440 - 580 - 640 = 1220

The two blanks add to 1220.

3Share the 1220 using the 140 gap

#7 Identify Subproblems 6.EE.B.7

If Q2 were as big as Q4, the pair would total 140 more, that is 1360, split evenly between them.

(1220 + 140) \div 2 = 680

Q4 is 680, so Q2 is 540.

Answer: 680 phones

4 · Reviewdoes it hold up?

Check both conditions: 580 + 540 + 640 + 680 = 2440, and 2440 divided by 4 is 610. Also 680 - 540 = 140.

Another way: Call Q4 the unknown: Q2 is that minus 140, so twice the unknown minus 140 equals 1220, giving 680 again.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading the average of 610 as a total of 2440 across four quarters.
6.SP.B.5 Summarize numerical data sets in relation to their context — Relating the table's known entries to the summary figure.
6.EE.B.7 Solve real-world problems by writing and solving equations of the form x + p = q — Splitting 1220 into two parts that differ by 140.

💡Takeaway. An average is a total in disguise: multiply it back out, and a blank in a table becomes an ordinary missing-number problem.

Where next?

← Back to the problem: Set up one-unknown equation from mean