Problem
Area of overlapping figures
A square with 7 cm sides overlaps a square with 10 cm sides. The overlapping area is never given. In each square the part outside the overlap is shaded. Find the difference between the two shaded areas.
Reasoning · Grade 5-1 Area of Figures
Your answer
Watch out The overlap seems essential, but in the difference it cancels out.
How to solve
Strategy Change Focus / Count the Complement — The one thing the problem refuses to tell me is the size of the overlap, which is a strong hint that it is not needed. So rather than hunting for it, I label the three regions on the drawing — small-only, shared, big-only — and notice that each shaded part is its own square minus the very same shared rectangle. Comparing the two shaded parts therefore means the shared rectangle is taken off both sides of the comparison and cancels out. To be certain the cancelling is real and not wishful thinking, I finish by guessing two different overlap sizes and checking that the difference comes out the same both times.
1STEP 1
Name the three regions on the picture
Split the picture into small part, overlap, large part.
Naming each region on the figure keeps track of what has been counted once and what has been counted twice, which is the only bookkeeping this problem needs.
3.MD.C.5Draw A Diagram2STEP 2
Find the area of each whole square
The two squares are 49 and 100.
7 × 7 = 49 cm², 10 × 10 = 100 cm²
A square is a rectangle whose two sides are equal, so the familiar length-times-width rule gives its area straight away.
3.MD.C.7Draw A Diagram3STEP 3
Write each shaded part as a square minus the overlap
Each shaded part is a square minus the overlap.
small shaded = 49 - overlap, large shaded = 100 - overlap
The area of a whole shape is the sum of the areas of the pieces it is cut into, so taking off the shared piece leaves exactly the shaded piece.
3.MD.C.7Change Focus Count The Complement4STEP 4
Take the difference and watch the overlap cancel
In the difference the overlap cancels.
(100 - overlap) - (49 - overlap) = 100 - 49 = 51 cm²
If two amounts each shrink by the same number, the gap between them is untouched — the same reason two people who each spend the same amount stay just as far apart in savings.
4.OA.A.3Change Focus Count The ComplementTaking the difference of the two shaded parts makes the overlap cancel, so its size never has to be found.
Why?
The same overlap is taken from each square, so subtracting the two results leaves the gap between the squares untouched.
🧱Shift both by the same amount and the difference holdsSlide two points the same distance along the line and the gap between them never changes.
Why?
Each shaded part is its square with the overlap removed, so the two descriptions can be set against each other exactly.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
5STEP 5
Test the cancelling with two guessed overlaps
So the difference is 100 − 49 = 51 cm².
92 - 41 = 51, 85 - 34 = 51
Trying particular numbers is a quick way to see that an unknown really has no effect, and the arithmetic is simple enough to do twice in your head.
3.NBT.A.2Guess And CheckAnswer
51 cm²
100 − 49 = 51
The answer is an area, so square centimetres are correct. It must also be less than 100 square centimetres, the whole big square, and it is. The sign is right too: the 10 cm square is the larger one, so its uncovered part is the bigger of the two and the difference is positive. And 51 is exactly the difference between the two whole squares, 100 minus 49, which is where the reasoning says it should land — the white rectangle, whatever its size, was taken from both parts equally and so left the gap alone.
Takeaway
Take the same amount away from two numbers and the gap between them never changes — so the overlap you cannot measure does not matter at all!
- Name the three regions on the picture
- Find the area of each whole square
- Write each shaded part as a square minus the overlap
- Take the difference and watch the overlap cancel
- Test the cancelling with two guessed overlaps