← Find original number range from rounded result · Pin Down a Number from Digit and Range Conditions

Find original number range from rounded result · 12 practice problems

5.NBT.A.43.NBT.A.1

Generated variants — 12

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

Variant 1 easy answer: at least 281 and at most 300

For the fifth graders at Munho's school to go on a field trip, they need at least 1515 buses, where each bus seats 2020 people.

What is the range of the number of fifth graders at Munho's school? Give the smallest and largest possible number ("at least ___ and at most ___").

Show solution
1 · Understandwhat's really being asked

The school needs 15 buses of 20 seats, and not fewer. That does not pin down the number of students -- it only says the group is too big for 14 buses and still fits in 15.

Givens
  • Each bus holds 20 people.
  • At least 15 buses are needed.
Unknowns
  • The smallest and largest possible number of fifth graders.
Constraints
  • The last bus does not have to be full, but 14 buses must not be enough.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check#8 Analyze the Units

Work out what 14 buses hold and what 15 hold. The group has to be above the first and no more than the second, which fixes both ends.

3 · Execute4 carry out the plan

1How many people fit in 14 buses?

#11 Work Backwards 5.NBT.A.4
If 14 buses were enough, 15 would not be needed -- so start by seeing what they hold.
14×20=28014 \times 20 = 280
280 students would fit in 14 buses.

2Smallest possible number of students

#6 Guess and Check 3.NBT.A.1
The group must be too big for that, so it takes just one more than 280.
280+1=281280 + 1 = 281
At least 281 students.

3How many people fit in 15 buses?

#8 Analyze the Units 5.NBT.A.4
All 15 buses full is the most the group could be.
15×20=30015 \times 20 = 300
300 seats altogether.

4Largest possible number of students

#6 Guess and Check 3.NBT.A.1
One student more than 300 would need a 16th bus.
281≤students≤300281 \le \text{students} \le 300
At most 300 students.
Answer: at least 281 and at most 300
4 · Reviewdoes it hold up?

Test both ends: 281 students fill 14 buses and spill one into the next, and 300 fills all 15 exactly. 280 would not need the extra bus, and 301 would need another.

Another way: Divide instead: 281 divided by 20 is 14 with a remainder, and 300 divided by 20 is exactly 15 -- the same two ends.

Standardsmin grade 5
  • 5.NBT.A.4 Use place value understanding to round decimals to any place. — Working out what 14 and 15 buses hold.
  • 3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100. — Reading 'at least this many buses' back into a range of group sizes.
💡Takeaway. A rounded-up count hides a range. One student over a full bus forces the next one, so the smallest group is just one past the load below.
Variant 2 easy answer: at least 276 and at most 300

For the fifth graders at Munho's school to go on a field trip, they need at least 1212 buses, where each bus seats 2525 people.

What is the range of the number of fifth graders at Munho's school? Give the smallest and largest possible number ("at least ___ and at most ___").

Show solution
1 · Understandwhat's really being asked

The school needs 12 buses of 25 seats, and not fewer. That does not pin down the number of students -- it only says the group is too big for 11 buses and still fits in 12.

Givens
  • Each bus holds 25 people.
  • At least 12 buses are needed.
Unknowns
  • The smallest and largest possible number of fifth graders.
Constraints
  • The last bus does not have to be full, but 11 buses must not be enough.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check#8 Analyze the Units

Work out what 11 buses hold and what 12 hold. The group has to be above the first and no more than the second, which fixes both ends.

3 · Execute4 carry out the plan

1How many people fit in 11 buses?

#11 Work Backwards 5.NBT.A.4
If 11 buses were enough, 12 would not be needed -- so start by seeing what they hold.
11×25=27511 \times 25 = 275
275 students would fit in 11 buses.

2Smallest possible number of students

#6 Guess and Check 3.NBT.A.1
The group must be too big for that, so it takes just one more than 275.
275+1=276275 + 1 = 276
At least 276 students.

3How many people fit in 12 buses?

#8 Analyze the Units 5.NBT.A.4
All 12 buses full is the most the group could be.
12×25=30012 \times 25 = 300
300 seats altogether.

4Largest possible number of students

#6 Guess and Check 3.NBT.A.1
One student more than 300 would need a 13th bus.
276≤students≤300276 \le \text{students} \le 300
At most 300 students.
Answer: at least 276 and at most 300
4 · Reviewdoes it hold up?

Test both ends: 276 students fill 11 buses and spill one into the next, and 300 fills all 12 exactly. 275 would not need the extra bus, and 301 would need another.

Another way: Divide instead: 276 divided by 25 is 11 with a remainder, and 300 divided by 25 is exactly 12 -- the same two ends.

Standardsmin grade 5
  • 5.NBT.A.4 Use place value understanding to round decimals to any place. — Working out what 11 and 12 buses hold.
  • 3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100. — Reading 'at least this many buses' back into a range of group sizes.
💡Takeaway. A rounded-up count hides a range. One student over a full bus forces the next one, so the smallest group is just one past the load below.
Variant 3 easy answer: at least 337 and at most 364

For the fifth graders at Munho's school to go on a field trip, they need at least 1313 buses, where each bus seats 2828 people.

What is the range of the number of fifth graders at Munho's school? Give the smallest and largest possible number ("at least ___ and at most ___").

Show solution
1 · Understandwhat's really being asked

The school needs 13 buses of 28 seats, and not fewer. That does not pin down the number of students -- it only says the group is too big for 12 buses and still fits in 13.

Givens
  • Each bus holds 28 people.
  • At least 13 buses are needed.
Unknowns
  • The smallest and largest possible number of fifth graders.
Constraints
  • The last bus does not have to be full, but 12 buses must not be enough.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check#8 Analyze the Units

Work out what 12 buses hold and what 13 hold. The group has to be above the first and no more than the second, which fixes both ends.

3 · Execute4 carry out the plan

1How many people fit in 12 buses?

#11 Work Backwards 5.NBT.A.4
If 12 buses were enough, 13 would not be needed -- so start by seeing what they hold.
12×28=33612 \times 28 = 336
336 students would fit in 12 buses.

2Smallest possible number of students

#6 Guess and Check 3.NBT.A.1
The group must be too big for that, so it takes just one more than 336.
336+1=337336 + 1 = 337
At least 337 students.

3How many people fit in 13 buses?

#8 Analyze the Units 5.NBT.A.4
All 13 buses full is the most the group could be.
13×28=36413 \times 28 = 364
364 seats altogether.

4Largest possible number of students

#6 Guess and Check 3.NBT.A.1
One student more than 364 would need a 14th bus.
337≤students≤364337 \le \text{students} \le 364
At most 364 students.
Answer: at least 337 and at most 364
4 · Reviewdoes it hold up?

Test both ends: 337 students fill 12 buses and spill one into the next, and 364 fills all 13 exactly. 336 would not need the extra bus, and 365 would need another.

Another way: Divide instead: 337 divided by 28 is 12 with a remainder, and 364 divided by 28 is exactly 13 -- the same two ends.

Standardsmin grade 5
  • 5.NBT.A.4 Use place value understanding to round decimals to any place. — Working out what 12 and 13 buses hold.
  • 3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100. — Reading 'at least this many buses' back into a range of group sizes.
💡Takeaway. A rounded-up count hides a range. One student over a full bus forces the next one, so the smallest group is just one past the load below.
Variant 4 easy answer: at least 241 and at most 270

For the fifth graders at Munho's school to go on a field trip, they need at least 99 buses, where each bus seats 3030 people.

What is the range of the number of fifth graders at Munho's school? Give the smallest and largest possible number ("at least ___ and at most ___").

Show solution
1 · Understandwhat's really being asked

The school needs 9 buses of 30 seats, and not fewer. That does not pin down the number of students -- it only says the group is too big for 8 buses and still fits in 9.

Givens
  • Each bus holds 30 people.
  • At least 9 buses are needed.
Unknowns
  • The smallest and largest possible number of fifth graders.
Constraints
  • The last bus does not have to be full, but 8 buses must not be enough.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check#8 Analyze the Units

Work out what 8 buses hold and what 9 hold. The group has to be above the first and no more than the second, which fixes both ends.

3 · Execute4 carry out the plan

1How many people fit in 8 buses?

#11 Work Backwards 5.NBT.A.4
If 8 buses were enough, 9 would not be needed -- so start by seeing what they hold.
8×30=2408 \times 30 = 240
240 students would fit in 8 buses.

2Smallest possible number of students

#6 Guess and Check 3.NBT.A.1
The group must be too big for that, so it takes just one more than 240.
240+1=241240 + 1 = 241
At least 241 students.

3How many people fit in 9 buses?

#8 Analyze the Units 5.NBT.A.4
All 9 buses full is the most the group could be.
9×30=2709 \times 30 = 270
270 seats altogether.

4Largest possible number of students

#6 Guess and Check 3.NBT.A.1
One student more than 270 would need a 10th bus.
241≤students≤270241 \le \text{students} \le 270
At most 270 students.
Answer: at least 241 and at most 270
4 · Reviewdoes it hold up?

Test both ends: 241 students fill 8 buses and spill one into the next, and 270 fills all 9 exactly. 240 would not need the extra bus, and 271 would need another.

Another way: Divide instead: 241 divided by 30 is 8 with a remainder, and 270 divided by 30 is exactly 9 -- the same two ends.

Standardsmin grade 5
  • 5.NBT.A.4 Use place value understanding to round decimals to any place. — Working out what 8 and 9 buses hold.
  • 3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100. — Reading 'at least this many buses' back into a range of group sizes.
💡Takeaway. A rounded-up count hides a range. One student over a full bus forces the next one, so the smallest group is just one past the load below.
Variant 5 medium answer: at least 321 and at most 352

For the fifth graders at Munho's school to go on a field trip, they need at least 1111 buses, where each bus seats 3232 people.

What is the range of the number of fifth graders at Munho's school? Give the smallest and largest possible number ("at least ___ and at most ___").

Show solution
1 · Understandwhat's really being asked

The school needs 11 buses of 32 seats, and not fewer. That does not pin down the number of students -- it only says the group is too big for 10 buses and still fits in 11.

Givens
  • Each bus holds 32 people.
  • At least 11 buses are needed.
Unknowns
  • The smallest and largest possible number of fifth graders.
Constraints
  • The last bus does not have to be full, but 10 buses must not be enough.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check#8 Analyze the Units

Work out what 10 buses hold and what 11 hold. The group has to be above the first and no more than the second, which fixes both ends.

3 · Execute4 carry out the plan

1How many people fit in 10 buses?

#11 Work Backwards 5.NBT.A.4
If 10 buses were enough, 11 would not be needed -- so start by seeing what they hold.
10×32=32010 \times 32 = 320
320 students would fit in 10 buses.

2Smallest possible number of students

#6 Guess and Check 3.NBT.A.1
The group must be too big for that, so it takes just one more than 320.
320+1=321320 + 1 = 321
At least 321 students.

3How many people fit in 11 buses?

#8 Analyze the Units 5.NBT.A.4
All 11 buses full is the most the group could be.
11×32=35211 \times 32 = 352
352 seats altogether.

4Largest possible number of students

#6 Guess and Check 3.NBT.A.1
One student more than 352 would need a 12th bus.
321≤students≤352321 \le \text{students} \le 352
At most 352 students.
Answer: at least 321 and at most 352
4 · Reviewdoes it hold up?

Test both ends: 321 students fill 10 buses and spill one into the next, and 352 fills all 11 exactly. 320 would not need the extra bus, and 353 would need another.

Another way: Divide instead: 321 divided by 32 is 10 with a remainder, and 352 divided by 32 is exactly 11 -- the same two ends.

Standardsmin grade 5
  • 5.NBT.A.4 Use place value understanding to round decimals to any place. — Working out what 10 and 11 buses hold.
  • 3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100. — Reading 'at least this many buses' back into a range of group sizes.
💡Takeaway. A rounded-up count hides a range. One student over a full bus forces the next one, so the smallest group is just one past the load below.
Variant 6 medium answer: at least 253 and at most 288

For the fifth graders at Munho's school to go on a field trip, they need at least 88 buses, where each bus seats 3636 people.

What is the range of the number of fifth graders at Munho's school? Give the smallest and largest possible number ("at least ___ and at most ___").

Show solution
1 · Understandwhat's really being asked

The school needs 8 buses of 36 seats, and not fewer. That does not pin down the number of students -- it only says the group is too big for 7 buses and still fits in 8.

Givens
  • Each bus holds 36 people.
  • At least 8 buses are needed.
Unknowns
  • The smallest and largest possible number of fifth graders.
Constraints
  • The last bus does not have to be full, but 7 buses must not be enough.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check#8 Analyze the Units

Work out what 7 buses hold and what 8 hold. The group has to be above the first and no more than the second, which fixes both ends.

3 · Execute4 carry out the plan

1How many people fit in 7 buses?

#11 Work Backwards 5.NBT.A.4
If 7 buses were enough, 8 would not be needed -- so start by seeing what they hold.
7×36=2527 \times 36 = 252
252 students would fit in 7 buses.

2Smallest possible number of students

#6 Guess and Check 3.NBT.A.1
The group must be too big for that, so it takes just one more than 252.
252+1=253252 + 1 = 253
At least 253 students.

3How many people fit in 8 buses?

#8 Analyze the Units 5.NBT.A.4
All 8 buses full is the most the group could be.
8×36=2888 \times 36 = 288
288 seats altogether.

4Largest possible number of students

#6 Guess and Check 3.NBT.A.1
One student more than 288 would need a 9th bus.
253≤students≤288253 \le \text{students} \le 288
At most 288 students.
Answer: at least 253 and at most 288
4 · Reviewdoes it hold up?

Test both ends: 253 students fill 7 buses and spill one into the next, and 288 fills all 8 exactly. 252 would not need the extra bus, and 289 would need another.

Another way: Divide instead: 253 divided by 36 is 7 with a remainder, and 288 divided by 36 is exactly 8 -- the same two ends.

Standardsmin grade 5
  • 5.NBT.A.4 Use place value understanding to round decimals to any place. — Working out what 7 and 8 buses hold.
  • 3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100. — Reading 'at least this many buses' back into a range of group sizes.
💡Takeaway. A rounded-up count hides a range. One student over a full bus forces the next one, so the smallest group is just one past the load below.
Variant 7 medium answer: at least 241 and at most 280

For the fifth graders at Munho's school to go on a field trip, they need at least 77 buses, where each bus seats 4040 people.

What is the range of the number of fifth graders at Munho's school? Give the smallest and largest possible number ("at least ___ and at most ___").

Show solution
1 · Understandwhat's really being asked

The school needs 7 buses of 40 seats, and not fewer. That does not pin down the number of students -- it only says the group is too big for 6 buses and still fits in 7.

Givens
  • Each bus holds 40 people.
  • At least 7 buses are needed.
Unknowns
  • The smallest and largest possible number of fifth graders.
Constraints
  • The last bus does not have to be full, but 6 buses must not be enough.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check#8 Analyze the Units

Work out what 6 buses hold and what 7 hold. The group has to be above the first and no more than the second, which fixes both ends.

3 · Execute4 carry out the plan

1How many people fit in 6 buses?

#11 Work Backwards 5.NBT.A.4
If 6 buses were enough, 7 would not be needed -- so start by seeing what they hold.
6×40=2406 \times 40 = 240
240 students would fit in 6 buses.

2Smallest possible number of students

#6 Guess and Check 3.NBT.A.1
The group must be too big for that, so it takes just one more than 240.
240+1=241240 + 1 = 241
At least 241 students.

3How many people fit in 7 buses?

#8 Analyze the Units 5.NBT.A.4
All 7 buses full is the most the group could be.
7×40=2807 \times 40 = 280
280 seats altogether.

4Largest possible number of students

#6 Guess and Check 3.NBT.A.1
One student more than 280 would need a 8th bus.
241≤students≤280241 \le \text{students} \le 280
At most 280 students.
Answer: at least 241 and at most 280
4 · Reviewdoes it hold up?

Test both ends: 241 students fill 6 buses and spill one into the next, and 280 fills all 7 exactly. 240 would not need the extra bus, and 281 would need another.

Another way: Divide instead: 241 divided by 40 is 6 with a remainder, and 280 divided by 40 is exactly 7 -- the same two ends.

Standardsmin grade 5
  • 5.NBT.A.4 Use place value understanding to round decimals to any place. — Working out what 6 and 7 buses hold.
  • 3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100. — Reading 'at least this many buses' back into a range of group sizes.
💡Takeaway. A rounded-up count hides a range. One student over a full bus forces the next one, so the smallest group is just one past the load below.
Variant 8 medium answer: at least 379 and at most 420

For the fifth graders at Munho's school to go on a field trip, they need at least 1010 buses, where each bus seats 4242 people.

What is the range of the number of fifth graders at Munho's school? Give the smallest and largest possible number ("at least ___ and at most ___").

Show solution
1 · Understandwhat's really being asked

The school needs 10 buses of 42 seats, and not fewer. That does not pin down the number of students -- it only says the group is too big for 9 buses and still fits in 10.

Givens
  • Each bus holds 42 people.
  • At least 10 buses are needed.
Unknowns
  • The smallest and largest possible number of fifth graders.
Constraints
  • The last bus does not have to be full, but 9 buses must not be enough.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check#8 Analyze the Units

Work out what 9 buses hold and what 10 hold. The group has to be above the first and no more than the second, which fixes both ends.

3 · Execute4 carry out the plan

1How many people fit in 9 buses?

#11 Work Backwards 5.NBT.A.4
If 9 buses were enough, 10 would not be needed -- so start by seeing what they hold.
9×42=3789 \times 42 = 378
378 students would fit in 9 buses.

2Smallest possible number of students

#6 Guess and Check 3.NBT.A.1
The group must be too big for that, so it takes just one more than 378.
378+1=379378 + 1 = 379
At least 379 students.

3How many people fit in 10 buses?

#8 Analyze the Units 5.NBT.A.4
All 10 buses full is the most the group could be.
10×42=42010 \times 42 = 420
420 seats altogether.

4Largest possible number of students

#6 Guess and Check 3.NBT.A.1
One student more than 420 would need a 11th bus.
379≤students≤420379 \le \text{students} \le 420
At most 420 students.
Answer: at least 379 and at most 420
4 · Reviewdoes it hold up?

Test both ends: 379 students fill 9 buses and spill one into the next, and 420 fills all 10 exactly. 378 would not need the extra bus, and 421 would need another.

Another way: Divide instead: 379 divided by 42 is 9 with a remainder, and 420 divided by 42 is exactly 10 -- the same two ends.

Standardsmin grade 5
  • 5.NBT.A.4 Use place value understanding to round decimals to any place. — Working out what 9 and 10 buses hold.
  • 3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100. — Reading 'at least this many buses' back into a range of group sizes.
💡Takeaway. A rounded-up count hides a range. One student over a full bus forces the next one, so the smallest group is just one past the load below.
Variant 9 hard answer: at least 181 and at most 225

For the fifth graders at Munho's school to go on a field trip, they need at least 55 buses, where each bus seats 4545 people.

What is the range of the number of fifth graders at Munho's school? Give the smallest and largest possible number ("at least ___ and at most ___").

Show solution
1 · Understandwhat's really being asked

The school needs 5 buses of 45 seats, and not fewer. That does not pin down the number of students -- it only says the group is too big for 4 buses and still fits in 5.

Givens
  • Each bus holds 45 people.
  • At least 5 buses are needed.
Unknowns
  • The smallest and largest possible number of fifth graders.
Constraints
  • The last bus does not have to be full, but 4 buses must not be enough.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check#8 Analyze the Units

Work out what 4 buses hold and what 5 hold. The group has to be above the first and no more than the second, which fixes both ends.

3 · Execute4 carry out the plan

1How many people fit in 4 buses?

#11 Work Backwards 5.NBT.A.4
If 4 buses were enough, 5 would not be needed -- so start by seeing what they hold.
4×45=1804 \times 45 = 180
180 students would fit in 4 buses.

2Smallest possible number of students

#6 Guess and Check 3.NBT.A.1
The group must be too big for that, so it takes just one more than 180.
180+1=181180 + 1 = 181
At least 181 students.

3How many people fit in 5 buses?

#8 Analyze the Units 5.NBT.A.4
All 5 buses full is the most the group could be.
5×45=2255 \times 45 = 225
225 seats altogether.

4Largest possible number of students

#6 Guess and Check 3.NBT.A.1
One student more than 225 would need a 6th bus.
181≤students≤225181 \le \text{students} \le 225
At most 225 students.
Answer: at least 181 and at most 225
4 · Reviewdoes it hold up?

Test both ends: 181 students fill 4 buses and spill one into the next, and 225 fills all 5 exactly. 180 would not need the extra bus, and 226 would need another.

Another way: Divide instead: 181 divided by 45 is 4 with a remainder, and 225 divided by 45 is exactly 5 -- the same two ends.

Standardsmin grade 5
  • 5.NBT.A.4 Use place value understanding to round decimals to any place. — Working out what 4 and 5 buses hold.
  • 3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100. — Reading 'at least this many buses' back into a range of group sizes.
💡Takeaway. A rounded-up count hides a range. One student over a full bus forces the next one, so the smallest group is just one past the load below.
Variant 10 hard answer: at least 241 and at most 288

For the fifth graders at Munho's school to go on a field trip, they need at least 66 buses, where each bus seats 4848 people.

What is the range of the number of fifth graders at Munho's school? Give the smallest and largest possible number ("at least ___ and at most ___").

Show solution
1 · Understandwhat's really being asked

The school needs 6 buses of 48 seats, and not fewer. That does not pin down the number of students -- it only says the group is too big for 5 buses and still fits in 6.

Givens
  • Each bus holds 48 people.
  • At least 6 buses are needed.
Unknowns
  • The smallest and largest possible number of fifth graders.
Constraints
  • The last bus does not have to be full, but 5 buses must not be enough.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check#8 Analyze the Units

Work out what 5 buses hold and what 6 hold. The group has to be above the first and no more than the second, which fixes both ends.

3 · Execute4 carry out the plan

1How many people fit in 5 buses?

#11 Work Backwards 5.NBT.A.4
If 5 buses were enough, 6 would not be needed -- so start by seeing what they hold.
5×48=2405 \times 48 = 240
240 students would fit in 5 buses.

2Smallest possible number of students

#6 Guess and Check 3.NBT.A.1
The group must be too big for that, so it takes just one more than 240.
240+1=241240 + 1 = 241
At least 241 students.

3How many people fit in 6 buses?

#8 Analyze the Units 5.NBT.A.4
All 6 buses full is the most the group could be.
6×48=2886 \times 48 = 288
288 seats altogether.

4Largest possible number of students

#6 Guess and Check 3.NBT.A.1
One student more than 288 would need a 7th bus.
241≤students≤288241 \le \text{students} \le 288
At most 288 students.
Answer: at least 241 and at most 288
4 · Reviewdoes it hold up?

Test both ends: 241 students fill 5 buses and spill one into the next, and 288 fills all 6 exactly. 240 would not need the extra bus, and 289 would need another.

Another way: Divide instead: 241 divided by 48 is 5 with a remainder, and 288 divided by 48 is exactly 6 -- the same two ends.

Standardsmin grade 5
  • 5.NBT.A.4 Use place value understanding to round decimals to any place. — Working out what 5 and 6 buses hold.
  • 3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100. — Reading 'at least this many buses' back into a range of group sizes.
💡Takeaway. A rounded-up count hides a range. One student over a full bus forces the next one, so the smallest group is just one past the load below.
Variant 11 hard answer: at least 166 and at most 220

For the fifth graders at Munho's school to go on a field trip, they need at least 44 buses, where each bus seats 5555 people.

What is the range of the number of fifth graders at Munho's school? Give the smallest and largest possible number ("at least ___ and at most ___").

Show solution
1 · Understandwhat's really being asked

The school needs 4 buses of 55 seats, and not fewer. That does not pin down the number of students -- it only says the group is too big for 3 buses and still fits in 4.

Givens
  • Each bus holds 55 people.
  • At least 4 buses are needed.
Unknowns
  • The smallest and largest possible number of fifth graders.
Constraints
  • The last bus does not have to be full, but 3 buses must not be enough.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check#8 Analyze the Units

Work out what 3 buses hold and what 4 hold. The group has to be above the first and no more than the second, which fixes both ends.

3 · Execute4 carry out the plan

1How many people fit in 3 buses?

#11 Work Backwards 5.NBT.A.4
If 3 buses were enough, 4 would not be needed -- so start by seeing what they hold.
3×55=1653 \times 55 = 165
165 students would fit in 3 buses.

2Smallest possible number of students

#6 Guess and Check 3.NBT.A.1
The group must be too big for that, so it takes just one more than 165.
165+1=166165 + 1 = 166
At least 166 students.

3How many people fit in 4 buses?

#8 Analyze the Units 5.NBT.A.4
All 4 buses full is the most the group could be.
4×55=2204 \times 55 = 220
220 seats altogether.

4Largest possible number of students

#6 Guess and Check 3.NBT.A.1
One student more than 220 would need a 5th bus.
166≤students≤220166 \le \text{students} \le 220
At most 220 students.
Answer: at least 166 and at most 220
4 · Reviewdoes it hold up?

Test both ends: 166 students fill 3 buses and spill one into the next, and 220 fills all 4 exactly. 165 would not need the extra bus, and 221 would need another.

Another way: Divide instead: 166 divided by 55 is 3 with a remainder, and 220 divided by 55 is exactly 4 -- the same two ends.

Standardsmin grade 5
  • 5.NBT.A.4 Use place value understanding to round decimals to any place. — Working out what 3 and 4 buses hold.
  • 3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100. — Reading 'at least this many buses' back into a range of group sizes.
💡Takeaway. A rounded-up count hides a range. One student over a full bus forces the next one, so the smallest group is just one past the load below.
Variant 12 hard answer: at least 121 and at most 180

For the fifth graders at Munho's school to go on a field trip, they need at least 33 buses, where each bus seats 6060 people.

What is the range of the number of fifth graders at Munho's school? Give the smallest and largest possible number ("at least ___ and at most ___").

Show solution
1 · Understandwhat's really being asked

The school needs 3 buses of 60 seats, and not fewer. That does not pin down the number of students -- it only says the group is too big for 2 buses and still fits in 3.

Givens
  • Each bus holds 60 people.
  • At least 3 buses are needed.
Unknowns
  • The smallest and largest possible number of fifth graders.
Constraints
  • The last bus does not have to be full, but 2 buses must not be enough.
2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check#8 Analyze the Units

Work out what 2 buses hold and what 3 hold. The group has to be above the first and no more than the second, which fixes both ends.

3 · Execute4 carry out the plan

1How many people fit in 2 buses?

#11 Work Backwards 5.NBT.A.4
If 2 buses were enough, 3 would not be needed -- so start by seeing what they hold.
2×60=1202 \times 60 = 120
120 students would fit in 2 buses.

2Smallest possible number of students

#6 Guess and Check 3.NBT.A.1
The group must be too big for that, so it takes just one more than 120.
120+1=121120 + 1 = 121
At least 121 students.

3How many people fit in 3 buses?

#8 Analyze the Units 5.NBT.A.4
All 3 buses full is the most the group could be.
3×60=1803 \times 60 = 180
180 seats altogether.

4Largest possible number of students

#6 Guess and Check 3.NBT.A.1
One student more than 180 would need a 4th bus.
121≤students≤180121 \le \text{students} \le 180
At most 180 students.
Answer: at least 121 and at most 180
4 · Reviewdoes it hold up?

Test both ends: 121 students fill 2 buses and spill one into the next, and 180 fills all 3 exactly. 120 would not need the extra bus, and 181 would need another.

Another way: Divide instead: 121 divided by 60 is 2 with a remainder, and 180 divided by 60 is exactly 3 -- the same two ends.

Standardsmin grade 5
  • 5.NBT.A.4 Use place value understanding to round decimals to any place. — Working out what 2 and 3 buses hold.
  • 3.NBT.A.1 Use place value understanding to round whole numbers to the nearest 10 or 100. — Reading 'at least this many buses' back into a range of group sizes.
💡Takeaway. A rounded-up count hides a range. One student over a full bus forces the next one, so the smallest group is just one past the load below.