Problem
How perimeter and area are related
A square has an unknown side length. Its width is pushed out 4 inches and its height up 2 inches. That adds 50 square inches of area. Find the area of the original square.
Reasoning · Grade 5-1 Perimeter and Area
Your answer
Watch out Finding the side 7 feels done, but the question wants the area.
How to solve
Strategy Draw a Diagram — The words describe a picture, so I draw it: the original square with an extra strip added along one side and another along the top. Drawn that way the added region is not a mysterious lump — it splits neatly into three rectangles, and finding the area of each is a separate easy subproblem. Two of those three depend on the unknown side, and the third is completely known, so the 50 square inches pins the side length down. I then guess-and-check the finished number back through the original wording to make sure the strip really does come to 50.
1STEP 1
Draw the square and the two strips
Draw the square with a side strip and a top strip.
new width = side + 4, new height = side + 2
Once the stretch is drawn instead of described, the added part is plainly made of flat rectangles, and rectangles are shapes whose areas a young solver already knows how to find.
3.MD.C.7Draw A Diagram2STEP 2
Split the added area into three easy pieces
The gain is two strips and a corner.
4 × side + 2 × side + 4 × 2
Breaking one awkward shape into three rectangles turns a hard area into three easy ones that simply add together, and the corner piece is the only one whose size is already fully known.
3.MD.C.7Identify SubproblemsThe added area splits into three plain rectangles whose areas are easy to write down.
Why?
The three pieces cover the added region with no gap and no overlap, so their areas add back to the whole gain.
🧱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?
Two of the three share the square's side as a factor, so that shared side can be pulled out and the sum written compactly.
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
3STEP 3
Add the pieces and use the 50
Since the total is 50, six sides make 42.
6 × side + 8 = 50 → 6 × side = 42
Peeling off the one piece whose area is fully known leaves a plain 'six of something makes 42' statement, which is far simpler than the picture it came from.
3.MD.C.7Identify Subproblems4STEP 4
Find the side length
The side is 7.
side = 42 ÷ 6 = 7 in
Finding the missing number in 'six times what makes 42' is a division fact from the multiplication table, so no algebra is needed to unlock the side length.
3.OA.A.4Identify Subproblems5STEP 5
Check the number against the original wording
The enlarged rectangle differs by 50 — it checks.
7 × 7 = 49, 11 × 9 = 99, 99 - 49 = 50
Running the candidate answer back through the original sentence, rather than through the working, catches any mistake made in setting the working up in the first place.
4.MD.A.3Guess And Check6STEP 6
State the area asked for
The area asked for is 49.
7 × 7 = 49 in²
It is easy to stop at the side length 7 once it appears, so reading the question again at the end makes sure the reported number is the one that was asked for.
4.MD.A.3Guess And CheckAnswer
49 square inches
7 × 7 = 49
The answer is an area, so square inches is the right unit, and 49 in squared corresponds to a 7-inch square, a sensible everyday size. The new rectangle is 11 by 9 = 99 square inches, so the shape roughly doubles in area, which matches an added strip of 50 in squared being about as big as the original 49 in squared. The added area also has to be more than 4 x 7 + 2 x 7 = 42 in squared, because of the corner piece, and 50 is comfortably more, by exactly the corner's 8 in squared. Nothing in the check is off by a stray strip or a missing corner.
Takeaway
Draw the stretched shape and the extra part splits into two strips and one corner — subtract the corner you already know, and the missing side length falls right out.
- Draw the square and the two strips
- Split the added area into three easy pieces
- Add the pieces and use the 50
- Find the side length
- Check the number against the original wording
- State the area asked for