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Use a partial average to find an unknown value · 12 practice problems

6.SP.A.36.SP.B.54.OA.A.3

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Generated variants — 12

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Variant 1 easy

The table shows the number of tourists who visited a town each month. The number of tourists in August was $72$ more than the average number of tourists over the other $3$ months. Find the average number of tourists per month over the $4$ months.

Month	June	July	August	September
Tourists	144	168		156

Your answer

Show solution

1 · Understandwhat's really being asked

Three months are on the table -- 144, 168 and 156 tourists. August is blank, but we are told it was 72 more than the average of those three. We need the average across all four months.

Givens

June 144, July 168, September 156.
August is 72 more than the average of the other three months.

Unknowns

The average over all four months.

Constraints

The clue compares August to the three-month average, not the four-month one.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#6 Guess and Check

The clue is about a three-month average, so build that first. It gives August, August gives the four-month total, and the total gives the average asked for.

3 · Execute4 carry out the plan

1Average of the three known months

#7 Identify Subproblems 6.SP.A.3

Add the three months on the table and share the total over three.

(144 + 168 + 156) \div 3 = 468 \div 3 = 156

Those three months averaged 156 tourists.

2Find August by working forward from the partial average

#11 Work Backwards 6.SP.B.5

August is 72 more than that average.

156 + 72 = 228

August had 228 tourists.

3Total all four months

#7 Identify Subproblems 4.OA.A.3

Add August to the three-month total.

468 + 228 = 696

696 tourists across the four months.

4Average over all four months

#6 Guess and Check 6.SP.A.3

Share the four-month total over four.

696 \div 4 = 174

The four-month average is 174 tourists.

Answer: 174 tourists

4 · Reviewdoes it hold up?

The four-month average 174 sits above the three-month average 156, which it must, since August was the biggest month by 72.

Another way: August pulls the average up by its excess shared over four months: 156 + 72 / 4 = 174, the same answer in one step.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading each average as a total shared evenly.
6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Using the three-month summary to recover the missing month.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Totalling all four months before averaging.

💡Takeaway. Check which numbers an average is over. The average of three months and the average of four are different amounts.

Variant 2 easy

The table shows the number of tourists who visited a town each month. The number of tourists in August was $84$ more than the average number of tourists over the other $3$ months. Find the average number of tourists per month over the $4$ months.

Month	June	July	August	September
Tourists	162	189		174

Your answer

Show solution

1 · Understandwhat's really being asked

Three months are on the table -- 162, 189 and 174 tourists. August is blank, but we are told it was 84 more than the average of those three. We need the average across all four months.

Givens

June 162, July 189, September 174.
August is 84 more than the average of the other three months.

Unknowns

The average over all four months.

Constraints

The clue compares August to the three-month average, not the four-month one.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#6 Guess and Check

The clue is about a three-month average, so build that first. It gives August, August gives the four-month total, and the total gives the average asked for.

3 · Execute4 carry out the plan

1Average of the three known months

#7 Identify Subproblems 6.SP.A.3

Add the three months on the table and share the total over three.

(162 + 189 + 174) \div 3 = 525 \div 3 = 175

Those three months averaged 175 tourists.

2Find August by working forward from the partial average

#11 Work Backwards 6.SP.B.5

August is 84 more than that average.

175 + 84 = 259

August had 259 tourists.

3Total all four months

#7 Identify Subproblems 4.OA.A.3

Add August to the three-month total.

525 + 259 = 784

784 tourists across the four months.

4Average over all four months

#6 Guess and Check 6.SP.A.3

Share the four-month total over four.

784 \div 4 = 196

The four-month average is 196 tourists.

Answer: 196 tourists

4 · Reviewdoes it hold up?

The four-month average 196 sits above the three-month average 175, which it must, since August was the biggest month by 84.

Another way: August pulls the average up by its excess shared over four months: 175 + 84 / 4 = 196, the same answer in one step.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading each average as a total shared evenly.
6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Using the three-month summary to recover the missing month.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Totalling all four months before averaging.

💡Takeaway. Check which numbers an average is over. The average of three months and the average of four are different amounts.

Variant 3 easy

The table shows the number of tourists who visited a town each month. The number of tourists in August was $96$ more than the average number of tourists over the other $3$ months. Find the average number of tourists per month over the $4$ months.

Month	June	July	August	September
Tourists	195	240		210

Your answer

Show solution

1 · Understandwhat's really being asked

Three months are on the table -- 195, 240 and 210 tourists. August is blank, but we are told it was 96 more than the average of those three. We need the average across all four months.

Givens

June 195, July 240, September 210.
August is 96 more than the average of the other three months.

Unknowns

The average over all four months.

Constraints

The clue compares August to the three-month average, not the four-month one.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#6 Guess and Check

The clue is about a three-month average, so build that first. It gives August, August gives the four-month total, and the total gives the average asked for.

3 · Execute4 carry out the plan

1Average of the three known months

#7 Identify Subproblems 6.SP.A.3

Add the three months on the table and share the total over three.

(195 + 240 + 210) \div 3 = 645 \div 3 = 215

Those three months averaged 215 tourists.

2Find August by working forward from the partial average

#11 Work Backwards 6.SP.B.5

August is 96 more than that average.

215 + 96 = 311

August had 311 tourists.

3Total all four months

#7 Identify Subproblems 4.OA.A.3

Add August to the three-month total.

645 + 311 = 956

956 tourists across the four months.

4Average over all four months

#6 Guess and Check 6.SP.A.3

Share the four-month total over four.

956 \div 4 = 239

The four-month average is 239 tourists.

Answer: 239 tourists

4 · Reviewdoes it hold up?

The four-month average 239 sits above the three-month average 215, which it must, since August was the biggest month by 96.

Another way: August pulls the average up by its excess shared over four months: 215 + 96 / 4 = 239, the same answer in one step.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading each average as a total shared evenly.
6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Using the three-month summary to recover the missing month.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Totalling all four months before averaging.

💡Takeaway. Check which numbers an average is over. The average of three months and the average of four are different amounts.

Variant 4 easy

The table shows the number of tourists who visited a town each month. The number of tourists in August was $108$ more than the average number of tourists over the other $3$ months. Find the average number of tourists per month over the $4$ months.

Month	June	July	August	September
Tourists	225	270		249

Your answer

Show solution

1 · Understandwhat's really being asked

Three months are on the table -- 225, 270 and 249 tourists. August is blank, but we are told it was 108 more than the average of those three. We need the average across all four months.

Givens

June 225, July 270, September 249.
August is 108 more than the average of the other three months.

Unknowns

The average over all four months.

Constraints

The clue compares August to the three-month average, not the four-month one.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#6 Guess and Check

The clue is about a three-month average, so build that first. It gives August, August gives the four-month total, and the total gives the average asked for.

3 · Execute4 carry out the plan

1Average of the three known months

#7 Identify Subproblems 6.SP.A.3

Add the three months on the table and share the total over three.

(225 + 270 + 249) \div 3 = 744 \div 3 = 248

Those three months averaged 248 tourists.

2Find August by working forward from the partial average

#11 Work Backwards 6.SP.B.5

August is 108 more than that average.

248 + 108 = 356

August had 356 tourists.

3Total all four months

#7 Identify Subproblems 4.OA.A.3

Add August to the three-month total.

744 + 356 = 1100

1100 tourists across the four months.

4Average over all four months

#6 Guess and Check 6.SP.A.3

Share the four-month total over four.

1100 \div 4 = 275

The four-month average is 275 tourists.

Answer: 275 tourists

4 · Reviewdoes it hold up?

The four-month average 275 sits above the three-month average 248, which it must, since August was the biggest month by 108.

Another way: August pulls the average up by its excess shared over four months: 248 + 108 / 4 = 275, the same answer in one step.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading each average as a total shared evenly.
6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Using the three-month summary to recover the missing month.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Totalling all four months before averaging.

💡Takeaway. Check which numbers an average is over. The average of three months and the average of four are different amounts.

Variant 5 medium

The table shows the number of tourists who visited a town each month. The number of tourists in August was $132$ more than the average number of tourists over the other $3$ months. Find the average number of tourists per month over the $4$ months.

Month	June	July	August	September
Tourists	288	312		297

Your answer

Show solution

1 · Understandwhat's really being asked

Three months are on the table -- 288, 312 and 297 tourists. August is blank, but we are told it was 132 more than the average of those three. We need the average across all four months.

Givens

June 288, July 312, September 297.
August is 132 more than the average of the other three months.

Unknowns

The average over all four months.

Constraints

The clue compares August to the three-month average, not the four-month one.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#6 Guess and Check

The clue is about a three-month average, so build that first. It gives August, August gives the four-month total, and the total gives the average asked for.

3 · Execute4 carry out the plan

1Average of the three known months

#7 Identify Subproblems 6.SP.A.3

Add the three months on the table and share the total over three.

(288 + 312 + 297) \div 3 = 897 \div 3 = 299

Those three months averaged 299 tourists.

2Find August by working forward from the partial average

#11 Work Backwards 6.SP.B.5

August is 132 more than that average.

299 + 132 = 431

August had 431 tourists.

3Total all four months

#7 Identify Subproblems 4.OA.A.3

Add August to the three-month total.

897 + 431 = 1328

1328 tourists across the four months.

4Average over all four months

#6 Guess and Check 6.SP.A.3

Share the four-month total over four.

1328 \div 4 = 332

The four-month average is 332 tourists.

Answer: 332 tourists

4 · Reviewdoes it hold up?

The four-month average 332 sits above the three-month average 299, which it must, since August was the biggest month by 132.

Another way: August pulls the average up by its excess shared over four months: 299 + 132 / 4 = 332, the same answer in one step.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading each average as a total shared evenly.
6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Using the three-month summary to recover the missing month.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Totalling all four months before averaging.

💡Takeaway. Check which numbers an average is over. The average of three months and the average of four are different amounts.

Variant 6 medium

The table shows the number of tourists who visited a town each month. The number of tourists in August was $168$ more than the average number of tourists over the other $3$ months. Find the average number of tourists per month over the $4$ months.

Month	June	July	August	September
Tourists	258	316		167

Your answer

Show solution

1 · Understandwhat's really being asked

Three months are on the table -- 258, 316 and 167 tourists. August is blank, but we are told it was 168 more than the average of those three. We need the average across all four months.

Givens

June 258, July 316, September 167.
August is 168 more than the average of the other three months.

Unknowns

The average over all four months.

Constraints

The clue compares August to the three-month average, not the four-month one.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#6 Guess and Check

The clue is about a three-month average, so build that first. It gives August, August gives the four-month total, and the total gives the average asked for.

3 · Execute4 carry out the plan

1Average of the three known months

#7 Identify Subproblems 6.SP.A.3

Add the three months on the table and share the total over three.

(258 + 316 + 167) \div 3 = 741 \div 3 = 247

Those three months averaged 247 tourists.

2Find August by working forward from the partial average

#11 Work Backwards 6.SP.B.5

August is 168 more than that average.

247 + 168 = 415

August had 415 tourists.

3Total all four months

#7 Identify Subproblems 4.OA.A.3

Add August to the three-month total.

741 + 415 = 1156

1156 tourists across the four months.

4Average over all four months

#6 Guess and Check 6.SP.A.3

Share the four-month total over four.

1156 \div 4 = 289

The four-month average is 289 tourists.

Answer: 289 tourists

4 · Reviewdoes it hold up?

The four-month average 289 sits above the three-month average 247, which it must, since August was the biggest month by 168.

Another way: August pulls the average up by its excess shared over four months: 247 + 168 / 4 = 289, the same answer in one step.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading each average as a total shared evenly.
6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Using the three-month summary to recover the missing month.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Totalling all four months before averaging.

💡Takeaway. Check which numbers an average is over. The average of three months and the average of four are different amounts.

Variant 7 medium

The table shows the number of tourists who visited a town each month. The number of tourists in August was $180$ more than the average number of tourists over the other $3$ months. Find the average number of tourists per month over the $4$ months.

Month	June	July	August	September
Tourists	330	285		300

Your answer

Show solution

1 · Understandwhat's really being asked

Three months are on the table -- 330, 285 and 300 tourists. August is blank, but we are told it was 180 more than the average of those three. We need the average across all four months.

Givens

June 330, July 285, September 300.
August is 180 more than the average of the other three months.

Unknowns

The average over all four months.

Constraints

The clue compares August to the three-month average, not the four-month one.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#6 Guess and Check

The clue is about a three-month average, so build that first. It gives August, August gives the four-month total, and the total gives the average asked for.

3 · Execute4 carry out the plan

1Average of the three known months

#7 Identify Subproblems 6.SP.A.3

Add the three months on the table and share the total over three.

(330 + 285 + 300) \div 3 = 915 \div 3 = 305

Those three months averaged 305 tourists.

2Find August by working forward from the partial average

#11 Work Backwards 6.SP.B.5

August is 180 more than that average.

305 + 180 = 485

August had 485 tourists.

3Total all four months

#7 Identify Subproblems 4.OA.A.3

Add August to the three-month total.

915 + 485 = 1400

1400 tourists across the four months.

4Average over all four months

#6 Guess and Check 6.SP.A.3

Share the four-month total over four.

1400 \div 4 = 350

The four-month average is 350 tourists.

Answer: 350 tourists

4 · Reviewdoes it hold up?

The four-month average 350 sits above the three-month average 305, which it must, since August was the biggest month by 180.

Another way: August pulls the average up by its excess shared over four months: 305 + 180 / 4 = 350, the same answer in one step.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading each average as a total shared evenly.
6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Using the three-month summary to recover the missing month.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Totalling all four months before averaging.

💡Takeaway. Check which numbers an average is over. The average of three months and the average of four are different amounts.

Variant 8 medium

The table shows the number of tourists who visited a town each month. The number of tourists in August was $160$ more than the average number of tourists over the other $3$ months. Find the average number of tourists per month over the $4$ months.

Month	June	July	August	September
Tourists	405	375		390

Your answer

Show solution

1 · Understandwhat's really being asked

Three months are on the table -- 405, 375 and 390 tourists. August is blank, but we are told it was 160 more than the average of those three. We need the average across all four months.

Givens

June 405, July 375, September 390.
August is 160 more than the average of the other three months.

Unknowns

The average over all four months.

Constraints

The clue compares August to the three-month average, not the four-month one.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#6 Guess and Check

The clue is about a three-month average, so build that first. It gives August, August gives the four-month total, and the total gives the average asked for.

3 · Execute4 carry out the plan

1Average of the three known months

#7 Identify Subproblems 6.SP.A.3

Add the three months on the table and share the total over three.

(405 + 375 + 390) \div 3 = 1170 \div 3 = 390

Those three months averaged 390 tourists.

2Find August by working forward from the partial average

#11 Work Backwards 6.SP.B.5

August is 160 more than that average.

390 + 160 = 550

August had 550 tourists.

3Total all four months

#7 Identify Subproblems 4.OA.A.3

Add August to the three-month total.

1170 + 550 = 1720

1720 tourists across the four months.

4Average over all four months

#6 Guess and Check 6.SP.A.3

Share the four-month total over four.

1720 \div 4 = 430

The four-month average is 430 tourists.

Answer: 430 tourists

4 · Reviewdoes it hold up?

The four-month average 430 sits above the three-month average 390, which it must, since August was the biggest month by 160.

Another way: August pulls the average up by its excess shared over four months: 390 + 160 / 4 = 430, the same answer in one step.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading each average as a total shared evenly.
6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Using the three-month summary to recover the missing month.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Totalling all four months before averaging.

💡Takeaway. Check which numbers an average is over. The average of three months and the average of four are different amounts.

Variant 9 hard

The table shows the number of tourists who visited a town each month. The number of tourists in August was $120$ more than the average number of tourists over the other $3$ months. Find the average number of tourists per month over the $4$ months.

Month	June	July	August	September
Tourists	410	380		350

Your answer

Show solution

1 · Understandwhat's really being asked

Three months are on the table -- 410, 380 and 350 tourists. August is blank, but we are told it was 120 more than the average of those three. We need the average across all four months.

Givens

June 410, July 380, September 350.
August is 120 more than the average of the other three months.

Unknowns

The average over all four months.

Constraints

The clue compares August to the three-month average, not the four-month one.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#6 Guess and Check

The clue is about a three-month average, so build that first. It gives August, August gives the four-month total, and the total gives the average asked for.

3 · Execute4 carry out the plan

1Average of the three known months

#7 Identify Subproblems 6.SP.A.3

Add the three months on the table and share the total over three.

(410 + 380 + 350) \div 3 = 1140 \div 3 = 380

Those three months averaged 380 tourists.

2Find August by working forward from the partial average

#11 Work Backwards 6.SP.B.5

August is 120 more than that average.

380 + 120 = 500

August had 500 tourists.

3Total all four months

#7 Identify Subproblems 4.OA.A.3

Add August to the three-month total.

1140 + 500 = 1640

1640 tourists across the four months.

4Average over all four months

#6 Guess and Check 6.SP.A.3

Share the four-month total over four.

1640 \div 4 = 410

The four-month average is 410 tourists.

Answer: 410 tourists

4 · Reviewdoes it hold up?

The four-month average 410 sits above the three-month average 380, which it must, since August was the biggest month by 120.

Another way: August pulls the average up by its excess shared over four months: 380 + 120 / 4 = 410, the same answer in one step.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading each average as a total shared evenly.
6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Using the three-month summary to recover the missing month.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Totalling all four months before averaging.

💡Takeaway. Check which numbers an average is over. The average of three months and the average of four are different amounts.

Variant 10 hard

The table shows the number of tourists who visited a town each month. The number of tourists in August was $200$ more than the average number of tourists over the other $3$ months. Find the average number of tourists per month over the $4$ months.

Month	June	July	August	September
Tourists	520	470		495

Your answer

Show solution

1 · Understandwhat's really being asked

Three months are on the table -- 520, 470 and 495 tourists. August is blank, but we are told it was 200 more than the average of those three. We need the average across all four months.

Givens

June 520, July 470, September 495.
August is 200 more than the average of the other three months.

Unknowns

The average over all four months.

Constraints

The clue compares August to the three-month average, not the four-month one.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#6 Guess and Check

The clue is about a three-month average, so build that first. It gives August, August gives the four-month total, and the total gives the average asked for.

3 · Execute4 carry out the plan

1Average of the three known months

#7 Identify Subproblems 6.SP.A.3

Add the three months on the table and share the total over three.

(520 + 470 + 495) \div 3 = 1485 \div 3 = 495

Those three months averaged 495 tourists.

2Find August by working forward from the partial average

#11 Work Backwards 6.SP.B.5

August is 200 more than that average.

495 + 200 = 695

August had 695 tourists.

3Total all four months

#7 Identify Subproblems 4.OA.A.3

Add August to the three-month total.

1485 + 695 = 2180

2180 tourists across the four months.

4Average over all four months

#6 Guess and Check 6.SP.A.3

Share the four-month total over four.

2180 \div 4 = 545

The four-month average is 545 tourists.

Answer: 545 tourists

4 · Reviewdoes it hold up?

The four-month average 545 sits above the three-month average 495, which it must, since August was the biggest month by 200.

Another way: August pulls the average up by its excess shared over four months: 495 + 200 / 4 = 545, the same answer in one step.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading each average as a total shared evenly.
6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Using the three-month summary to recover the missing month.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Totalling all four months before averaging.

💡Takeaway. Check which numbers an average is over. The average of three months and the average of four are different amounts.

Variant 11 hard

The table shows the number of tourists who visited a town each month. The number of tourists in August was $220$ more than the average number of tourists over the other $3$ months. Find the average number of tourists per month over the $4$ months.

Month	June	July	August	September
Tourists	540	510		525

Your answer

Show solution

1 · Understandwhat's really being asked

Three months are on the table -- 540, 510 and 525 tourists. August is blank, but we are told it was 220 more than the average of those three. We need the average across all four months.

Givens

June 540, July 510, September 525.
August is 220 more than the average of the other three months.

Unknowns

The average over all four months.

Constraints

The clue compares August to the three-month average, not the four-month one.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#6 Guess and Check

The clue is about a three-month average, so build that first. It gives August, August gives the four-month total, and the total gives the average asked for.

3 · Execute4 carry out the plan

1Average of the three known months

#7 Identify Subproblems 6.SP.A.3

Add the three months on the table and share the total over three.

(540 + 510 + 525) \div 3 = 1575 \div 3 = 525

Those three months averaged 525 tourists.

2Find August by working forward from the partial average

#11 Work Backwards 6.SP.B.5

August is 220 more than that average.

525 + 220 = 745

August had 745 tourists.

3Total all four months

#7 Identify Subproblems 4.OA.A.3

Add August to the three-month total.

1575 + 745 = 2320

2320 tourists across the four months.

4Average over all four months

#6 Guess and Check 6.SP.A.3

Share the four-month total over four.

2320 \div 4 = 580

The four-month average is 580 tourists.

Answer: 580 tourists

4 · Reviewdoes it hold up?

The four-month average 580 sits above the three-month average 525, which it must, since August was the biggest month by 220.

Another way: August pulls the average up by its excess shared over four months: 525 + 220 / 4 = 580, the same answer in one step.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading each average as a total shared evenly.
6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Using the three-month summary to recover the missing month.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Totalling all four months before averaging.

💡Takeaway. Check which numbers an average is over. The average of three months and the average of four are different amounts.

Variant 12 hard

The table shows the number of tourists who visited a town each month. The number of tourists in August was $240$ more than the average number of tourists over the other $3$ months. Find the average number of tourists per month over the $4$ months.

Month	June	July	August	September
Tourists	612	588		570

Your answer

Show solution

1 · Understandwhat's really being asked

Three months are on the table -- 612, 588 and 570 tourists. August is blank, but we are told it was 240 more than the average of those three. We need the average across all four months.

Givens

June 612, July 588, September 570.
August is 240 more than the average of the other three months.

Unknowns

The average over all four months.

Constraints

The clue compares August to the three-month average, not the four-month one.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #7 Identify Subproblems#6 Guess and Check

The clue is about a three-month average, so build that first. It gives August, August gives the four-month total, and the total gives the average asked for.

3 · Execute4 carry out the plan

1Average of the three known months

#7 Identify Subproblems 6.SP.A.3

Add the three months on the table and share the total over three.

(612 + 588 + 570) \div 3 = 1770 \div 3 = 590

Those three months averaged 590 tourists.

2Find August by working forward from the partial average

#11 Work Backwards 6.SP.B.5

August is 240 more than that average.

590 + 240 = 830

August had 830 tourists.

3Total all four months

#7 Identify Subproblems 4.OA.A.3

Add August to the three-month total.

1770 + 830 = 2600

2600 tourists across the four months.

4Average over all four months

#6 Guess and Check 6.SP.A.3

Share the four-month total over four.

2600 \div 4 = 650

The four-month average is 650 tourists.

Answer: 650 tourists

4 · Reviewdoes it hold up?

The four-month average 650 sits above the three-month average 590, which it must, since August was the biggest month by 240.

Another way: August pulls the average up by its excess shared over four months: 590 + 240 / 4 = 650, the same answer in one step.

Standardsmin grade 6

6.SP.A.3 Recognize that a measure of center summarizes all its values with a single number — Reading each average as a total shared evenly.
6.SP.B.5 Summarize numerical data sets by reporting number of observations and measures — Using the three-month summary to recover the missing month.
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Totalling all four months before averaging.

💡Takeaway. Check which numbers an average is over. The average of three months and the average of four are different amounts.

Where next?

← Back to the problem: Use a partial average to find an unknown value