Problem
Add or subtract equations to eliminate unknowns
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.
Expressions
Your answer
How to solve
Strategy Draw a Diagram — Draw the two weighings side by side as bars in a row plus a box. Lining them up shows that the only difference between the two pictures is the 2 extra bars, so the jump in weight must be exactly those 2 bars. That turns one hard question into two easy subproblems: first find one bar's weight, then subtract the bars from a total to leave the box.
1STEP 1
Picture the two weighings
Draw both weighings: the same box, once with 3 bars at 455 g and once with 5 bars at 605 g.
Seeing the same box in both rows makes it clear the box weight is shared and can be set aside.
4.OA.A.3Draw A Diagram2STEP 2
Compare to find the extra weight
The box cancels, so 605 - 455 = 150 g belongs to the 2 extra bars alone.
605 - 455 = 150
The box weight cancels when we compare the two totals, leaving only the part that changed.
4.NBT.B.4Draw A DiagramThe 150 g difference between the two weighings is the weight of exactly 2 soap bars.
Why?
Both weighings contain the very same box, so when the two totals are compared the box's weight sits on both sides and cancels out of the difference.
Why?
Each weighing is the box and its bars with nothing left out and nothing counted twice, so the two totals can differ only where the contents differ.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
The only change between the two weighings was putting in 2 more bars, so the difference measures just those 2 bars.
Why?
Taking the earlier total away from the later one undoes the adding that happened in between, leaving only what was put in.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
3STEP 3
Find one bar's weight
Those 150 g are the weight of 2 identical bars, so split it into 2 equal parts to find one bar.
150 ÷ 2 = 75
Equal bars means equal shares, so dividing the extra weight by 2 gives one bar.
4.OA.A.3Identify Subproblems4STEP 4
Find the box alone
Three bars weigh 3 x 75 = 225 g, so the box alone is 455 - 225 = 230 g.
3 × 75 = 225, 455 - 225 = 230
Once one bar is known, the box is simply the total minus all the soap inside it.
5.OA.A.2Identify SubproblemsAnswer
230 g
Check with the second weighing: box 230 g + 5 bars (5 x 75 = 375 g) = 605 g, which matches. The box (230 g) is heavier than one bar (75 g) but lighter than the full 455 g total, so the magnitude makes sense.
Takeaway
When two situations share the same hidden part, compare them - the difference tells you exactly what changed.
- Picture the two weighings
- Compare to find the extra weight
- Find one bar's weight
- Find the box alone
Where next?
Another one like thissuggested
Step back and solidifysuggested