Depth · one concept from first principles · step 2 of 4
A rate gives the area of a part
In rectangle ABCD at the right, the shaded part's area is of the whole area. What is the area of the shaded part, in ?
RatiosGeometry
Your answer
How to solve
Strategy Identify Subproblems — A rate is an instruction to multiply. Find the whole first, then apply the rate to it -- the shape of the shaded region never has to be worked out.
1STEP 1
Find the whole area
The rectangle's area is its two sides multiplied.
14 × 9 = 126
The rectangle covers 126 cm².
6.G.A.1Draw A Diagram2STEP 2
Apply the rate to the whole
The shaded part is 31/63 of that area, so multiply. Notice 126 is exactly twice 63, which makes the multiplication easy.
126 ÷ 63 = 2
Each sixty-third of the rectangle is 2 cm².
6.RP.A.3Analyze The UnitsOne sixty-third of the rectangle is .
Why?
A rate of measures the shaded part in sixty-thirds of the whole, so the whole is sixty-three equal pieces of that size.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
The rectangle covers , and sixty-three equal pieces of it are each.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
3STEP 3
Count 31 of those parts
The shaded region is 31 of those equal parts.
2 × 31 = 62
The shaded area is 62 cm².
6.RP.A.3Identify SubproblemsAnswer
62 cm²
Check the rate: 62 ÷ 126 reduces to 31/63, as given.
Takeaway
A rate is an instruction to multiply -- find the whole, then apply it.
- Find the whole area
- Apply the rate to the whole
- Count 31 of those parts
Where next?
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Step back and solidifysuggested
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