← Compare two weighings to cancel the container · Find Two Unknowns from Sum and Difference

Compare two weighings to cancel the container · 12 practice problems

4.OA.A.35.NBT.B.6

Generated variants — 12

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

Variant 1 easy answer: 150 g

A box holding $6$ identical bars of soap weighs $300\ \text{g}$ . After putting in $2$ more identical bars of soap, the box weighs $350\ \text{g}$ . How many grams does the empty box weigh by itself?

Show solution

1 · Understandwhat's really being asked

A box with 6 equal soap bars inside weighs 300 g. When 2 more equal bars are added, the same box now holds 8 bars and weighs 350 g. We need the weight of the empty box alone.

Givens

The box plus 6 bars weighs 300 g.
The box plus 8 bars weighs 350 g.
Every bar of soap weighs the same.

Unknowns

The weight of the empty box.

Constraints

The same box is used in both weighings, so only the bars changed.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw the two weighings side by side as a box plus a row of bars. Lining them up shows the only difference is the extra bars, so comparing the two totals cancels the box and leaves the bars alone.

3 · Execute4 carry out the plan

1Picture the two weighings

#1 Draw a Diagram 4.OA.A.3

Draw the first weighing as the box with 6 equal bars (300 g) and the second as the same box with 8 bars (350 g). The box is identical in both pictures.

\text{box} + 6\ \text{bars} = 300,\quad \text{box} + 8\ \text{bars} = 350

Seeing the same box in both rows makes it clear its weight is shared and can be set aside.

2Compare to find the extra weight

#1 Draw a Diagram 4.OA.A.3

The second weighing is heavier only because of the 2 extra bars, so subtracting gives the weight of just those bars.

350 - 300 = 50

The box weight cancels when the two totals are compared, leaving only what changed.

3Find one bar's weight

#7 Identify Subproblems 5.NBT.B.6

Those 50 g are the weight of 2 identical bars, so split it into 2 equal parts.

50 \div 2 = 25

Equal bars share the extra weight equally, so one bar is 25 g.

4Find the box alone

#7 Identify Subproblems 4.OA.A.3

In the first weighing, 300 g is the box plus 6 bars. Those 6 bars weigh 150 g, so removing them leaves the box.

6 \times 25 = 150,\quad 300 - 150 = 150

Once one bar is known, the box is the total minus all the soap inside it.

Answer: 150 g

4 · Reviewdoes it hold up?

Check both weighings: 150 + 6 x 25 = 300 and 150 + 8 x 25 = 350. Both match the numbers given.

Another way: Work from the second weighing instead: 350 - 200 = 150, which gives the same box weight.

Standardsmin grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Comparing two weighings and removing the known bars to reach the box weight.
5.NBT.B.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors — Dividing 50 by 2 to find the weight of one bar.

💡Takeaway. When the same thing sits in both weighings, compare them instead of solving each one: whatever is shared cancels out.

Variant 2 easy answer: 125 g

A box holding $5$ identical bars of soap weighs $300\ \text{g}$ . After putting in $2$ more identical bars of soap, the box weighs $370\ \text{g}$ . How many grams does the empty box weigh by itself?

Show solution

1 · Understandwhat's really being asked

A box with 5 equal soap bars inside weighs 300 g. When 2 more equal bars are added, the same box now holds 7 bars and weighs 370 g. We need the weight of the empty box alone.

Givens

The box plus 5 bars weighs 300 g.
The box plus 7 bars weighs 370 g.
Every bar of soap weighs the same.

Unknowns

The weight of the empty box.

Constraints

The same box is used in both weighings, so only the bars changed.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw the two weighings side by side as a box plus a row of bars. Lining them up shows the only difference is the extra bars, so comparing the two totals cancels the box and leaves the bars alone.

3 · Execute4 carry out the plan

1Picture the two weighings

#1 Draw a Diagram 4.OA.A.3

Draw the first weighing as the box with 5 equal bars (300 g) and the second as the same box with 7 bars (370 g). The box is identical in both pictures.

\text{box} + 5\ \text{bars} = 300,\quad \text{box} + 7\ \text{bars} = 370

Seeing the same box in both rows makes it clear its weight is shared and can be set aside.

2Compare to find the extra weight

#1 Draw a Diagram 4.OA.A.3

The second weighing is heavier only because of the 2 extra bars, so subtracting gives the weight of just those bars.

370 - 300 = 70

The box weight cancels when the two totals are compared, leaving only what changed.

3Find one bar's weight

#7 Identify Subproblems 5.NBT.B.6

Those 70 g are the weight of 2 identical bars, so split it into 2 equal parts.

70 \div 2 = 35

Equal bars share the extra weight equally, so one bar is 35 g.

4Find the box alone

#7 Identify Subproblems 4.OA.A.3

In the first weighing, 300 g is the box plus 5 bars. Those 5 bars weigh 175 g, so removing them leaves the box.

5 \times 35 = 175,\quad 300 - 175 = 125

Once one bar is known, the box is the total minus all the soap inside it.

Answer: 125 g

4 · Reviewdoes it hold up?

Check both weighings: 125 + 5 x 35 = 300 and 125 + 7 x 35 = 370. Both match the numbers given.

Another way: Work from the second weighing instead: 370 - 245 = 125, which gives the same box weight.

Standardsmin grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Comparing two weighings and removing the known bars to reach the box weight.
5.NBT.B.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors — Dividing 70 by 2 to find the weight of one bar.

💡Takeaway. When the same thing sits in both weighings, compare them instead of solving each one: whatever is shared cancels out.

Variant 3 easy answer: 140 g

A box holding $2$ identical bars of soap weighs $260\ \text{g}$ . After putting in $3$ more identical bars of soap, the box weighs $440\ \text{g}$ . How many grams does the empty box weigh by itself?

Show solution

1 · Understandwhat's really being asked

A box with 2 equal soap bars inside weighs 260 g. When 3 more equal bars are added, the same box now holds 5 bars and weighs 440 g. We need the weight of the empty box alone.

Givens

The box plus 2 bars weighs 260 g.
The box plus 5 bars weighs 440 g.
Every bar of soap weighs the same.

Unknowns

The weight of the empty box.

Constraints

The same box is used in both weighings, so only the bars changed.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw the two weighings side by side as a box plus a row of bars. Lining them up shows the only difference is the extra bars, so comparing the two totals cancels the box and leaves the bars alone.

3 · Execute4 carry out the plan

1Picture the two weighings

#1 Draw a Diagram 4.OA.A.3

Draw the first weighing as the box with 2 equal bars (260 g) and the second as the same box with 5 bars (440 g). The box is identical in both pictures.

\text{box} + 2\ \text{bars} = 260,\quad \text{box} + 5\ \text{bars} = 440

Seeing the same box in both rows makes it clear its weight is shared and can be set aside.

2Compare to find the extra weight

#1 Draw a Diagram 4.OA.A.3

The second weighing is heavier only because of the 3 extra bars, so subtracting gives the weight of just those bars.

440 - 260 = 180

The box weight cancels when the two totals are compared, leaving only what changed.

3Find one bar's weight

#7 Identify Subproblems 5.NBT.B.6

Those 180 g are the weight of 3 identical bars, so split it into 3 equal parts.

180 \div 3 = 60

Equal bars share the extra weight equally, so one bar is 60 g.

4Find the box alone

#7 Identify Subproblems 4.OA.A.3

In the first weighing, 260 g is the box plus 2 bars. Those 2 bars weigh 120 g, so removing them leaves the box.

2 \times 60 = 120,\quad 260 - 120 = 140

Once one bar is known, the box is the total minus all the soap inside it.

Answer: 140 g

4 · Reviewdoes it hold up?

Check both weighings: 140 + 2 x 60 = 260 and 140 + 5 x 60 = 440. Both match the numbers given.

Another way: Work from the second weighing instead: 440 - 300 = 140, which gives the same box weight.

Standardsmin grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Comparing two weighings and removing the known bars to reach the box weight.
5.NBT.B.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors — Dividing 180 by 3 to find the weight of one bar.

💡Takeaway. When the same thing sits in both weighings, compare them instead of solving each one: whatever is shared cancels out.

Variant 4 easy answer: 190 g

A box holding $4$ identical bars of soap weighs $370\ \text{g}$ . After putting in $2$ more identical bars of soap, the box weighs $460\ \text{g}$ . How many grams does the empty box weigh by itself?

Show solution

1 · Understandwhat's really being asked

A box with 4 equal soap bars inside weighs 370 g. When 2 more equal bars are added, the same box now holds 6 bars and weighs 460 g. We need the weight of the empty box alone.

Givens

The box plus 4 bars weighs 370 g.
The box plus 6 bars weighs 460 g.
Every bar of soap weighs the same.

Unknowns

The weight of the empty box.

Constraints

The same box is used in both weighings, so only the bars changed.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw the two weighings side by side as a box plus a row of bars. Lining them up shows the only difference is the extra bars, so comparing the two totals cancels the box and leaves the bars alone.

3 · Execute4 carry out the plan

1Picture the two weighings

#1 Draw a Diagram 4.OA.A.3

Draw the first weighing as the box with 4 equal bars (370 g) and the second as the same box with 6 bars (460 g). The box is identical in both pictures.

\text{box} + 4\ \text{bars} = 370,\quad \text{box} + 6\ \text{bars} = 460

Seeing the same box in both rows makes it clear its weight is shared and can be set aside.

2Compare to find the extra weight

#1 Draw a Diagram 4.OA.A.3

The second weighing is heavier only because of the 2 extra bars, so subtracting gives the weight of just those bars.

460 - 370 = 90

The box weight cancels when the two totals are compared, leaving only what changed.

3Find one bar's weight

#7 Identify Subproblems 5.NBT.B.6

Those 90 g are the weight of 2 identical bars, so split it into 2 equal parts.

90 \div 2 = 45

Equal bars share the extra weight equally, so one bar is 45 g.

4Find the box alone

#7 Identify Subproblems 4.OA.A.3

In the first weighing, 370 g is the box plus 4 bars. Those 4 bars weigh 180 g, so removing them leaves the box.

4 \times 45 = 180,\quad 370 - 180 = 190

Once one bar is known, the box is the total minus all the soap inside it.

Answer: 190 g

4 · Reviewdoes it hold up?

Check both weighings: 190 + 4 x 45 = 370 and 190 + 6 x 45 = 460. Both match the numbers given.

Another way: Work from the second weighing instead: 460 - 270 = 190, which gives the same box weight.

Standardsmin grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Comparing two weighings and removing the known bars to reach the box weight.
5.NBT.B.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors — Dividing 90 by 2 to find the weight of one bar.

💡Takeaway. When the same thing sits in both weighings, compare them instead of solving each one: whatever is shared cancels out.

Variant 5 medium answer: 195 g

A box holding $5$ identical bars of soap weighs $395\ \text{g}$ . After putting in $3$ more identical bars of soap, the box weighs $515\ \text{g}$ . How many grams does the empty box weigh by itself?

Show solution

1 · Understandwhat's really being asked

A box with 5 equal soap bars inside weighs 395 g. When 3 more equal bars are added, the same box now holds 8 bars and weighs 515 g. We need the weight of the empty box alone.

Givens

The box plus 5 bars weighs 395 g.
The box plus 8 bars weighs 515 g.
Every bar of soap weighs the same.

Unknowns

The weight of the empty box.

Constraints

The same box is used in both weighings, so only the bars changed.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw the two weighings side by side as a box plus a row of bars. Lining them up shows the only difference is the extra bars, so comparing the two totals cancels the box and leaves the bars alone.

3 · Execute4 carry out the plan

1Picture the two weighings

#1 Draw a Diagram 4.OA.A.3

Draw the first weighing as the box with 5 equal bars (395 g) and the second as the same box with 8 bars (515 g). The box is identical in both pictures.

\text{box} + 5\ \text{bars} = 395,\quad \text{box} + 8\ \text{bars} = 515

Seeing the same box in both rows makes it clear its weight is shared and can be set aside.

2Compare to find the extra weight

#1 Draw a Diagram 4.OA.A.3

The second weighing is heavier only because of the 3 extra bars, so subtracting gives the weight of just those bars.

515 - 395 = 120

The box weight cancels when the two totals are compared, leaving only what changed.

3Find one bar's weight

#7 Identify Subproblems 5.NBT.B.6

Those 120 g are the weight of 3 identical bars, so split it into 3 equal parts.

120 \div 3 = 40

Equal bars share the extra weight equally, so one bar is 40 g.

4Find the box alone

#7 Identify Subproblems 4.OA.A.3

In the first weighing, 395 g is the box plus 5 bars. Those 5 bars weigh 200 g, so removing them leaves the box.

5 \times 40 = 200,\quad 395 - 200 = 195

Once one bar is known, the box is the total minus all the soap inside it.

Answer: 195 g

4 · Reviewdoes it hold up?

Check both weighings: 195 + 5 x 40 = 395 and 195 + 8 x 40 = 515. Both match the numbers given.

Another way: Work from the second weighing instead: 515 - 320 = 195, which gives the same box weight.

Standardsmin grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Comparing two weighings and removing the known bars to reach the box weight.
5.NBT.B.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors — Dividing 120 by 3 to find the weight of one bar.

💡Takeaway. When the same thing sits in both weighings, compare them instead of solving each one: whatever is shared cancels out.

Variant 6 medium answer: 210 g

A box holding $2$ identical bars of soap weighs $320\ \text{g}$ . After putting in $4$ more identical bars of soap, the box weighs $540\ \text{g}$ . How many grams does the empty box weigh by itself?

Show solution

1 · Understandwhat's really being asked

A box with 2 equal soap bars inside weighs 320 g. When 4 more equal bars are added, the same box now holds 6 bars and weighs 540 g. We need the weight of the empty box alone.

Givens

The box plus 2 bars weighs 320 g.
The box plus 6 bars weighs 540 g.
Every bar of soap weighs the same.

Unknowns

The weight of the empty box.

Constraints

The same box is used in both weighings, so only the bars changed.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw the two weighings side by side as a box plus a row of bars. Lining them up shows the only difference is the extra bars, so comparing the two totals cancels the box and leaves the bars alone.

3 · Execute4 carry out the plan

1Picture the two weighings

#1 Draw a Diagram 4.OA.A.3

Draw the first weighing as the box with 2 equal bars (320 g) and the second as the same box with 6 bars (540 g). The box is identical in both pictures.

\text{box} + 2\ \text{bars} = 320,\quad \text{box} + 6\ \text{bars} = 540

Seeing the same box in both rows makes it clear its weight is shared and can be set aside.

2Compare to find the extra weight

#1 Draw a Diagram 4.OA.A.3

The second weighing is heavier only because of the 4 extra bars, so subtracting gives the weight of just those bars.

540 - 320 = 220

The box weight cancels when the two totals are compared, leaving only what changed.

3Find one bar's weight

#7 Identify Subproblems 5.NBT.B.6

Those 220 g are the weight of 4 identical bars, so split it into 4 equal parts.

220 \div 4 = 55

Equal bars share the extra weight equally, so one bar is 55 g.

4Find the box alone

#7 Identify Subproblems 4.OA.A.3

In the first weighing, 320 g is the box plus 2 bars. Those 2 bars weigh 110 g, so removing them leaves the box.

2 \times 55 = 110,\quad 320 - 110 = 210

Once one bar is known, the box is the total minus all the soap inside it.

Answer: 210 g

4 · Reviewdoes it hold up?

Check both weighings: 210 + 2 x 55 = 320 and 210 + 6 x 55 = 540. Both match the numbers given.

Another way: Work from the second weighing instead: 540 - 330 = 210, which gives the same box weight.

Standardsmin grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Comparing two weighings and removing the known bars to reach the box weight.
5.NBT.B.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors — Dividing 220 by 4 to find the weight of one bar.

💡Takeaway. When the same thing sits in both weighings, compare them instead of solving each one: whatever is shared cancels out.

Variant 7 medium answer: 230 g

A box holding $3$ identical bars of soap weighs $455\ \text{g}$ . After putting in $2$ more identical bars of soap, the box weighs $605\ \text{g}$ . How many grams does the empty box weigh by itself?

Show solution

1 · Understandwhat's really being asked

A box with 3 equal soap bars inside weighs 455 g. When 2 more equal bars are added, the same box now holds 5 bars and weighs 605 g. We need the weight of the empty box alone.

Givens

The box plus 3 bars weighs 455 g.
The box plus 5 bars weighs 605 g.
Every bar of soap weighs the same.

Unknowns

The weight of the empty box.

Constraints

The same box is used in both weighings, so only the bars changed.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw the two weighings side by side as a box plus a row of bars. Lining them up shows the only difference is the extra bars, so comparing the two totals cancels the box and leaves the bars alone.

3 · Execute4 carry out the plan

1Picture the two weighings

#1 Draw a Diagram 4.OA.A.3

Draw the first weighing as the box with 3 equal bars (455 g) and the second as the same box with 5 bars (605 g). The box is identical in both pictures.

\text{box} + 3\ \text{bars} = 455,\quad \text{box} + 5\ \text{bars} = 605

Seeing the same box in both rows makes it clear its weight is shared and can be set aside.

2Compare to find the extra weight

#1 Draw a Diagram 4.OA.A.3

The second weighing is heavier only because of the 2 extra bars, so subtracting gives the weight of just those bars.

605 - 455 = 150

The box weight cancels when the two totals are compared, leaving only what changed.

3Find one bar's weight

#7 Identify Subproblems 5.NBT.B.6

Those 150 g are the weight of 2 identical bars, so split it into 2 equal parts.

150 \div 2 = 75

Equal bars share the extra weight equally, so one bar is 75 g.

4Find the box alone

#7 Identify Subproblems 4.OA.A.3

In the first weighing, 455 g is the box plus 3 bars. Those 3 bars weigh 225 g, so removing them leaves the box.

3 \times 75 = 225,\quad 455 - 225 = 230

Once one bar is known, the box is the total minus all the soap inside it.

Answer: 230 g

4 · Reviewdoes it hold up?

Check both weighings: 230 + 3 x 75 = 455 and 230 + 5 x 75 = 605. Both match the numbers given.

Another way: Work from the second weighing instead: 605 - 375 = 230, which gives the same box weight.

Standardsmin grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Comparing two weighings and removing the known bars to reach the box weight.
5.NBT.B.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors — Dividing 150 by 2 to find the weight of one bar.

💡Takeaway. When the same thing sits in both weighings, compare them instead of solving each one: whatever is shared cancels out.

Variant 8 medium answer: 220 g

A box holding $4$ identical bars of soap weighs $420\ \text{g}$ . After putting in $4$ more identical bars of soap, the box weighs $620\ \text{g}$ . How many grams does the empty box weigh by itself?

Show solution

1 · Understandwhat's really being asked

A box with 4 equal soap bars inside weighs 420 g. When 4 more equal bars are added, the same box now holds 8 bars and weighs 620 g. We need the weight of the empty box alone.

Givens

The box plus 4 bars weighs 420 g.
The box plus 8 bars weighs 620 g.
Every bar of soap weighs the same.

Unknowns

The weight of the empty box.

Constraints

The same box is used in both weighings, so only the bars changed.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw the two weighings side by side as a box plus a row of bars. Lining them up shows the only difference is the extra bars, so comparing the two totals cancels the box and leaves the bars alone.

3 · Execute4 carry out the plan

1Picture the two weighings

#1 Draw a Diagram 4.OA.A.3

Draw the first weighing as the box with 4 equal bars (420 g) and the second as the same box with 8 bars (620 g). The box is identical in both pictures.

\text{box} + 4\ \text{bars} = 420,\quad \text{box} + 8\ \text{bars} = 620

Seeing the same box in both rows makes it clear its weight is shared and can be set aside.

2Compare to find the extra weight

#1 Draw a Diagram 4.OA.A.3

The second weighing is heavier only because of the 4 extra bars, so subtracting gives the weight of just those bars.

620 - 420 = 200

The box weight cancels when the two totals are compared, leaving only what changed.

3Find one bar's weight

#7 Identify Subproblems 5.NBT.B.6

Those 200 g are the weight of 4 identical bars, so split it into 4 equal parts.

200 \div 4 = 50

Equal bars share the extra weight equally, so one bar is 50 g.

4Find the box alone

#7 Identify Subproblems 4.OA.A.3

In the first weighing, 420 g is the box plus 4 bars. Those 4 bars weigh 200 g, so removing them leaves the box.

4 \times 50 = 200,\quad 420 - 200 = 220

Once one bar is known, the box is the total minus all the soap inside it.

Answer: 220 g

4 · Reviewdoes it hold up?

Check both weighings: 220 + 4 x 50 = 420 and 220 + 8 x 50 = 620. Both match the numbers given.

Another way: Work from the second weighing instead: 620 - 400 = 220, which gives the same box weight.

Standardsmin grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Comparing two weighings and removing the known bars to reach the box weight.
5.NBT.B.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors — Dividing 200 by 4 to find the weight of one bar.

💡Takeaway. When the same thing sits in both weighings, compare them instead of solving each one: whatever is shared cancels out.

Variant 9 hard answer: 175 g

A box holding $4$ identical bars of soap weighs $435\ \text{g}$ . After putting in $3$ more identical bars of soap, the box weighs $630\ \text{g}$ . How many grams does the empty box weigh by itself?

Show solution

1 · Understandwhat's really being asked

A box with 4 equal soap bars inside weighs 435 g. When 3 more equal bars are added, the same box now holds 7 bars and weighs 630 g. We need the weight of the empty box alone.

Givens

The box plus 4 bars weighs 435 g.
The box plus 7 bars weighs 630 g.
Every bar of soap weighs the same.

Unknowns

The weight of the empty box.

Constraints

The same box is used in both weighings, so only the bars changed.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw the two weighings side by side as a box plus a row of bars. Lining them up shows the only difference is the extra bars, so comparing the two totals cancels the box and leaves the bars alone.

3 · Execute4 carry out the plan

1Picture the two weighings

#1 Draw a Diagram 4.OA.A.3

Draw the first weighing as the box with 4 equal bars (435 g) and the second as the same box with 7 bars (630 g). The box is identical in both pictures.

\text{box} + 4\ \text{bars} = 435,\quad \text{box} + 7\ \text{bars} = 630

Seeing the same box in both rows makes it clear its weight is shared and can be set aside.

2Compare to find the extra weight

#1 Draw a Diagram 4.OA.A.3

The second weighing is heavier only because of the 3 extra bars, so subtracting gives the weight of just those bars.

630 - 435 = 195

The box weight cancels when the two totals are compared, leaving only what changed.

3Find one bar's weight

#7 Identify Subproblems 5.NBT.B.6

Those 195 g are the weight of 3 identical bars, so split it into 3 equal parts.

195 \div 3 = 65

Equal bars share the extra weight equally, so one bar is 65 g.

4Find the box alone

#7 Identify Subproblems 4.OA.A.3

In the first weighing, 435 g is the box plus 4 bars. Those 4 bars weigh 260 g, so removing them leaves the box.

4 \times 65 = 260,\quad 435 - 260 = 175

Once one bar is known, the box is the total minus all the soap inside it.

Answer: 175 g

4 · Reviewdoes it hold up?

Check both weighings: 175 + 4 x 65 = 435 and 175 + 7 x 65 = 630. Both match the numbers given.

Another way: Work from the second weighing instead: 630 - 455 = 175, which gives the same box weight.

Standardsmin grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Comparing two weighings and removing the known bars to reach the box weight.
5.NBT.B.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors — Dividing 195 by 3 to find the weight of one bar.

💡Takeaway. When the same thing sits in both weighings, compare them instead of solving each one: whatever is shared cancels out.

Variant 10 hard answer: 240 g

A box holding $2$ identical bars of soap weighs $480\ \text{g}$ . After putting in $2$ more identical bars of soap, the box weighs $720\ \text{g}$ . How many grams does the empty box weigh by itself?

Show solution

1 · Understandwhat's really being asked

A box with 2 equal soap bars inside weighs 480 g. When 2 more equal bars are added, the same box now holds 4 bars and weighs 720 g. We need the weight of the empty box alone.

Givens

The box plus 2 bars weighs 480 g.
The box plus 4 bars weighs 720 g.
Every bar of soap weighs the same.

Unknowns

The weight of the empty box.

Constraints

The same box is used in both weighings, so only the bars changed.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw the two weighings side by side as a box plus a row of bars. Lining them up shows the only difference is the extra bars, so comparing the two totals cancels the box and leaves the bars alone.

3 · Execute4 carry out the plan

1Picture the two weighings

#1 Draw a Diagram 4.OA.A.3

Draw the first weighing as the box with 2 equal bars (480 g) and the second as the same box with 4 bars (720 g). The box is identical in both pictures.

\text{box} + 2\ \text{bars} = 480,\quad \text{box} + 4\ \text{bars} = 720

Seeing the same box in both rows makes it clear its weight is shared and can be set aside.

2Compare to find the extra weight

#1 Draw a Diagram 4.OA.A.3

The second weighing is heavier only because of the 2 extra bars, so subtracting gives the weight of just those bars.

720 - 480 = 240

The box weight cancels when the two totals are compared, leaving only what changed.

3Find one bar's weight

#7 Identify Subproblems 5.NBT.B.6

Those 240 g are the weight of 2 identical bars, so split it into 2 equal parts.

240 \div 2 = 120

Equal bars share the extra weight equally, so one bar is 120 g.

4Find the box alone

#7 Identify Subproblems 4.OA.A.3

In the first weighing, 480 g is the box plus 2 bars. Those 2 bars weigh 240 g, so removing them leaves the box.

2 \times 120 = 240,\quad 480 - 240 = 240

Once one bar is known, the box is the total minus all the soap inside it.

Answer: 240 g

4 · Reviewdoes it hold up?

Check both weighings: 240 + 2 x 120 = 480 and 240 + 4 x 120 = 720. Both match the numbers given.

Another way: Work from the second weighing instead: 720 - 480 = 240, which gives the same box weight.

Standardsmin grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Comparing two weighings and removing the known bars to reach the box weight.
5.NBT.B.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors — Dividing 240 by 2 to find the weight of one bar.

💡Takeaway. When the same thing sits in both weighings, compare them instead of solving each one: whatever is shared cancels out.

Variant 11 hard answer: 260 g

A box holding $3$ identical bars of soap weighs $500\ \text{g}$ . After putting in $3$ more identical bars of soap, the box weighs $740\ \text{g}$ . How many grams does the empty box weigh by itself?

Show solution

1 · Understandwhat's really being asked

A box with 3 equal soap bars inside weighs 500 g. When 3 more equal bars are added, the same box now holds 6 bars and weighs 740 g. We need the weight of the empty box alone.

Givens

The box plus 3 bars weighs 500 g.
The box plus 6 bars weighs 740 g.
Every bar of soap weighs the same.

Unknowns

The weight of the empty box.

Constraints

The same box is used in both weighings, so only the bars changed.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw the two weighings side by side as a box plus a row of bars. Lining them up shows the only difference is the extra bars, so comparing the two totals cancels the box and leaves the bars alone.

3 · Execute4 carry out the plan

1Picture the two weighings

#1 Draw a Diagram 4.OA.A.3

Draw the first weighing as the box with 3 equal bars (500 g) and the second as the same box with 6 bars (740 g). The box is identical in both pictures.

\text{box} + 3\ \text{bars} = 500,\quad \text{box} + 6\ \text{bars} = 740

Seeing the same box in both rows makes it clear its weight is shared and can be set aside.

2Compare to find the extra weight

#1 Draw a Diagram 4.OA.A.3

The second weighing is heavier only because of the 3 extra bars, so subtracting gives the weight of just those bars.

740 - 500 = 240

The box weight cancels when the two totals are compared, leaving only what changed.

3Find one bar's weight

#7 Identify Subproblems 5.NBT.B.6

Those 240 g are the weight of 3 identical bars, so split it into 3 equal parts.

240 \div 3 = 80

Equal bars share the extra weight equally, so one bar is 80 g.

4Find the box alone

#7 Identify Subproblems 4.OA.A.3

In the first weighing, 500 g is the box plus 3 bars. Those 3 bars weigh 240 g, so removing them leaves the box.

3 \times 80 = 240,\quad 500 - 240 = 260

Once one bar is known, the box is the total minus all the soap inside it.

Answer: 260 g

4 · Reviewdoes it hold up?

Check both weighings: 260 + 3 x 80 = 500 and 260 + 6 x 80 = 740. Both match the numbers given.

Another way: Work from the second weighing instead: 740 - 480 = 260, which gives the same box weight.

Standardsmin grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Comparing two weighings and removing the known bars to reach the box weight.
5.NBT.B.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors — Dividing 240 by 3 to find the weight of one bar.

💡Takeaway. When the same thing sits in both weighings, compare them instead of solving each one: whatever is shared cancels out.

Variant 12 hard answer: 310 g

A box holding $3$ identical bars of soap weighs $580\ \text{g}$ . After putting in $2$ more identical bars of soap, the box weighs $760\ \text{g}$ . How many grams does the empty box weigh by itself?

Show solution

1 · Understandwhat's really being asked

A box with 3 equal soap bars inside weighs 580 g. When 2 more equal bars are added, the same box now holds 5 bars and weighs 760 g. We need the weight of the empty box alone.

Givens

The box plus 3 bars weighs 580 g.
The box plus 5 bars weighs 760 g.
Every bar of soap weighs the same.

Unknowns

The weight of the empty box.

Constraints

The same box is used in both weighings, so only the bars changed.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

Draw the two weighings side by side as a box plus a row of bars. Lining them up shows the only difference is the extra bars, so comparing the two totals cancels the box and leaves the bars alone.

3 · Execute4 carry out the plan

1Picture the two weighings

#1 Draw a Diagram 4.OA.A.3

Draw the first weighing as the box with 3 equal bars (580 g) and the second as the same box with 5 bars (760 g). The box is identical in both pictures.

\text{box} + 3\ \text{bars} = 580,\quad \text{box} + 5\ \text{bars} = 760

Seeing the same box in both rows makes it clear its weight is shared and can be set aside.

2Compare to find the extra weight

#1 Draw a Diagram 4.OA.A.3

The second weighing is heavier only because of the 2 extra bars, so subtracting gives the weight of just those bars.

760 - 580 = 180

The box weight cancels when the two totals are compared, leaving only what changed.

3Find one bar's weight

#7 Identify Subproblems 5.NBT.B.6

Those 180 g are the weight of 2 identical bars, so split it into 2 equal parts.

180 \div 2 = 90

Equal bars share the extra weight equally, so one bar is 90 g.

4Find the box alone

#7 Identify Subproblems 4.OA.A.3

In the first weighing, 580 g is the box plus 3 bars. Those 3 bars weigh 270 g, so removing them leaves the box.

3 \times 90 = 270,\quad 580 - 270 = 310

Once one bar is known, the box is the total minus all the soap inside it.

Answer: 310 g

4 · Reviewdoes it hold up?

Check both weighings: 310 + 3 x 90 = 580 and 310 + 5 x 90 = 760. Both match the numbers given.

Another way: Work from the second weighing instead: 760 - 450 = 310, which gives the same box weight.

Standardsmin grade 5

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers — Comparing two weighings and removing the known bars to reach the box weight.
5.NBT.B.6 Find whole-number quotients with up to four-digit dividends and two-digit divisors — Dividing 180 by 2 to find the weight of one bar.

💡Takeaway. When the same thing sits in both weighings, compare them instead of solving each one: whatever is shared cancels out.