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Math Expert V
Joined: 02 Sep 2009
Posts: 58347
A black and white drawing shows four circles of equal radius. Three of  [#permalink]

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Difficulty:   35% (medium)

Question Stats: 70% (02:08) correct 30% (03:05) wrong based on 56 sessions

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A black and white drawing shows four circles of equal radius. Three of the circles are each divided into k equal parts, where k > 5. The fourth circle is divided into k - 4 equal parts. A child colors one of the k parts in each of the three circles, and one of the k - 4 parts in the fourth circle. What part of a whole circle did the child color?

(A) $$\frac{4k - 12}{k(k - 4)}$$

(B) $$\frac{4k - 8}{k(k - 4)}$$

(C) $$\frac{4k - 4}{k(k - 4)}$$

(D) $$\frac{2k - 4}{k(k - 4)}$$

(E) $$\frac{k + 4}{k(k - 4)}$$

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A black and white drawing shows four circles of equal radius. Three of  [#permalink]

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Bunuel wrote:
A black and white drawing shows four circles of equal radius. Three of the circles are each divided into k equal parts, where k > 5. The fourth circle is divided into k - 4 equal parts. A child colors one of the k parts in each of the three circles, and one of the k - 4 parts in the fourth circle. What part of a whole circle did the child color?

(A) $$\frac{4k - 12}{k(k - 4)}$$

(B) $$\frac{4k - 8}{k(k - 4)}$$

(C) $$\frac{4k - 4}{k(k - 4)}$$

(D) $$\frac{2k - 4}{k(k - 4)}$$

(E) $$\frac{k + 4}{k(k - 4)}$$

Since 3 of the four circles have k parts and the fourth circle is divided into k-4 parts,
an easy approach to solving this problem is to fix an arbitrary value of k. Let k be 6.

Now, the child colors one of the 6 parts in three circles and one of the 2 parts in the 4 circles.

Therefore, $$3*\frac{1}{6} + \frac{1}{2} = \frac{1}{2} + \frac{1}{2} = 1$$ circle of the 4 circles is coloured

Evaluating answer options(using value of k = 6)

(A) $$\frac{4k - 12}{k(k - 4)}$$ = $$\frac{24 - 12}{6(6 - 4)} = 1$$

(B) $$\frac{4k - 8}{k(k - 4)}$$ = $$\frac{24 - 8}{6(6 - 4)} = \frac{4}{3}$$

(C) $$\frac{4k - 4}{k(k - 4)}$$ = $$\frac{24 - 4}{6(6 - 4)} = \frac{5}{3}$$

(D) $$\frac{2k - 4}{k(k - 4)}$$ = $$\frac{12 - 4}{6(6 - 4)} = \frac{2}{3}$$

(E) $$\frac{k + 4}{k(k - 4)}$$ = $$\frac{6 + 4}{6(6 - 4)} = \frac{5}{6}$$
(Option A)
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Senior SC Moderator V
Joined: 22 May 2016
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A black and white drawing shows four circles of equal radius. Three of  [#permalink]

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Bunuel wrote:
A black and white drawing shows four circles of equal radius. Three of the circles are each divided into k equal parts, where k > 5. The fourth circle is divided into k - 4 equal parts. A child colors one of the k parts in each of the three circles, and one of the k - 4 parts in the fourth circle. What part of a whole circle did the child color?

(A) $$\frac{4k - 12}{k(k - 4)}$$

(B) $$\frac{4k - 8}{k(k - 4)}$$

(C) $$\frac{4k - 4}{k(k - 4)}$$

(D) $$\frac{2k - 4}{k(k - 4)}$$

(E) $$\frac{k + 4}{k(k - 4)}$$

+1 kudos to pushpitkc , whose answer I did not see
while I was formalizing what I drew for this problem.
Attachment: circparts.png [ 19.23 KiB | Viewed 871 times ]

Use k = 6, and sketch (it's fast - a little over 1 min to solve)
(6-4) leaves more than one part for the last circle.

Draw four circles.

Three circles are divided into k = 6 parts
Divide three circles into sixths
Shade one part of each. Shaded part = $$\frac{1}{6}$$ of a whole circle

The fourth circle is divided into (k-4) = (6-4) = 2 parts
Divide the fourth circle in half. Shade one half.
The one shaded part = $$\frac{1}{2}=\frac{3}{6}$$ of a whole circle

The total of the shaded areas is
• 3 circles: $$(\frac{1}{6} + \frac{1}{6} + \frac{1}{6}) = \frac{3}{6}$$
• Fourth circle: $$\frac{1}{2} = \frac{3}{6}$$
• Combined: $$(\frac{3}{6} + \frac{3}{6}) = \frac{6}{6}= 1$$ whole circle

Using k = 6, find the answer that yields 1

(A) $$\frac{4k - 12}{k(k - 4)}$$
$$=\frac{(24-12)}{6(2)}=\frac{12}{12}=1$$ CORRECT

(B) $$\frac{4k - 8}{k(k - 4)}$$
=$$\frac{(24 - 8)}{6(2)} = \frac{16}{12}$$ REJECT

(C) $$\frac{4k - 4}{k(k - 4)}$$
=$$\frac{(24 - 4)}{6(2)} = \frac{20}{12}$$ REJECT

(D) $$\frac{2k - 4}{k(k - 4)}$$
=$$\frac{(12 - 4)}{6(2)} = \frac{8}{12}$$ REJECT

(E) $$\frac{k + 4}{k(k - 4)}$$
=$$\frac{(6 + 4)}{6(2)} = \frac{10}{12}$$ REJECT

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Joined: 01 Feb 2017
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A black and white drawing shows four circles of equal radius. Three of  [#permalink]

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1/k + 1/k + 1/k + 1/(k-4)=

3/k + 1/(k-4)=

{3(k-4)+k}/k(k-4)=

4k-12/k(k-4)

Ans A A black and white drawing shows four circles of equal radius. Three of   [#permalink] 24 Mar 2018, 01:31
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