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A certain cake recipe states that the cake should be baked i

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A certain cake recipe states that the cake should be baked i [#permalink] New post 22 Mar 2018, 03:13
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A certain cake recipe states that the cake should be baked in a pan 8 inches in diameter. If Jules wants to use the recipe to make a cake of the same depth but 12 inches in diameter, by what factor should he multiply the recipe ingredients?

(A) \(2 \frac{1}{2}\)

(B) \(2 \frac{1}{4}\)

(C) \(1 \frac{1}{2}\)

(D) \(1 \frac{4}{9}\)

(E) \(1 \frac{1}{3}\)
[Reveal] Spoiler: OA

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Re: A certain cake recipe states that the cake should be baked i [#permalink] New post 22 Mar 2018, 20:16
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One way to solve this problem is to find the actual areas of the 8 inch diameter pie, and the area of the 12 inch diameter pie, and divide the latter by the former. But this problem can be solved almost instantly if you know a very useful rule: if you multiply any length of one shape to get the length of a similar shape, then if you want the area of the latter shape, you can simply square the multiplier you just used. Here's a simple example. If you have two similar triangles, and the perimeter of the larger triangle is 3 times that of the smaller triangle, then you know the area is 3^2 times larger, or 9 times larger.

So, using this technique, when we see that the ratio of pie diameter is 12 to 8, or 3/2, we can simply square 3/2, getting 9/4, which is B.
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Re: A certain cake recipe states that the cake should be baked i [#permalink] New post 28 Jun 2020, 07:55
SherpaPrep wrote:
One way to solve this problem is to find the actual areas of the 8 inch diameter pie, and the area of the 12 inch diameter pie, and divide the latter by the former. But this problem can be solved almost instantly if you know a very useful rule: if you multiply any length of one shape to get the length of a similar shape, then if you want the area of the latter shape, you can simply square the multiplier you just used. Here's a simple example. If you have two similar triangles, and the perimeter of the larger triangle is 3 times that of the smaller triangle, then you know the area is 3^2 times larger, or 9 times larger.

So, using this technique, when we see that the ratio of pie diameter is 12 to 8, or 3/2, we can simply square 3/2, getting 9/4, which is B.



I didn't understand the rule.
May you explain it in a detailed way?
and squaring it is valid only for the total area or perimeter ?
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Re: A certain cake recipe states that the cake should be baked i [#permalink] New post 29 Jun 2020, 06:46
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Carcass wrote:
A certain cake recipe states that the cake should be baked in a pan 8 inches in diameter. If Jules wants to use the recipe to make a cake of the same depth but 12 inches in diameter, by what factor should he multiply the recipe ingredients?

(A) \(2 \frac{1}{2}\)

(B) \(2 \frac{1}{4}\)

(C) \(1 \frac{1}{2}\)

(D) \(1 \frac{4}{9}\)

(E) \(1 \frac{1}{3}\)



The main idea here is that the volume of ingredients needed is proportional to the volume of the cake.
For example, if the volume of the cake with diameter 12 is TWICE the volume of the cake with diameter 8, then we will need TWICE has many ingredients.

Although the question doesn't mention it, the shape of the cake is cylindrical.

Volume of cylinder \(= \pi r^2 h\)

ORIGINAL CAKE
Original cake recipe requires a pan with an 8-inch diameter.
If the diameter is 8 inches, then the RADIUS = 4 inches
Since we don't know the height of the cake, let's just use the variable \(h\) to represent the height
So, the volume \(= \pi (4^2)h\)
Simplify to get: \(= 16\pi h\)


DIFFERENT CAKE
This cake requires a pan with an 12-inch diameter, which means the RADIUS = 6 inches
So, the volume \(= \pi (6^2)h\)
Simplify to get: \(= 36\pi h\)


The question becomes, "\(= 36\pi h\) is how many times greater than \(= 16\pi h\)?"

To answer this, we must simplify: \(\frac{36\pi h}{16\pi h}\)

Cancel \(pi h\) to get: \(\frac{36}{16}\)

Simplify further to get: \(\frac{9}{4}\)

Rewrite as mixed fraction: \(2\frac{1}{4}\)

Answer: B

Cheers,
Brent
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Re: A certain cake recipe states that the cake should be baked i   [#permalink] 29 Jun 2020, 06:46
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