A black and white drawing shows four circles of equal radius. Three of

Question

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

(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)}

pushpitkc · Answer

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)

generis · Answer

+1 kudos to pushpitkc , whose answer I did not see while I was formalizing what I drew for this problem. circparts.png 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 Answer A

Shobhit7 · Answer

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

bumpbot · Answer

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