dhruv09arora wrote:

Machine X can produce 3 widgets every 2 minutes and Machine Y produces 5 widgets every 3 minutes. If both the machines work together for 25 minutes what part of the production will be finished by Machine X?

(A) 2/19

(B) 3/19

(C) 4/19

(D) 5/19

(E) 9/19

One approach: find (Part X : Part Y)

Then (Part X : Whole X+Y)

I. Change the time, find (part: part) ratio, then (part: whole) Use ANY easier time period to find PART : PART ratio

Relative portions that X and Y finish

will not change as long as time worked is the same

\(X\)'s part of total production is:

\(\frac{X}{X+Y}\)(1) Change the time period.

• X's output + Y's output = total output

• Total output for each is:

\(X\): \(r_{x}*T=W_{x}\)

\(Y\): \(r_{y}*T=W_{y}\)

Same time worked? Then number of widgets each makes depends ONLY on their rates.

(2): compare

rates with a time that is easier than 25 minutes

• Use LCM of 6 minutes to get an identical time period worked, which will yield

relative portions of total work each produces

\(X\)'s rate in 6 minutes: \(\frac{3w}{2min}=\frac{9w}{6min}\)

\(Y\)'s rate in 6 minutes: \(\frac{5w}{3min}=\frac{10w}{6min}\)

• The part of production finished by X?

\(\frac{X}{(X+Y)}=\frac{9}{9+10}=\frac{9}{19}\)Answer E II. Harder: TOTAL output in 25 minutes In 25 minutes, X finishes

\(\frac{X_{w}}{Total_{w}}\)• \(X\), total output:

\(W_{x}=r_{x}*25\)

\(W_{x}=(\frac{3w}{2min}*25min)=\frac{75}{2}w\)

• \(Y\), total output

\(W_{y}=r_{y}*25\)

\(W_{y}=(\frac{5w}{3min}*25min)=\frac{125}{3}w\)

• Overall total output (X + Y):

\((X+Y)=(\frac{75}{2}w+\frac{125}{3}w)=\)

\((\frac{225}{6w}+\frac{250w}{6}w)=\frac{475}{6}w\)

• X finishes what part of total production?\(\frac{X}{Total}=\frac{(\frac{75}{2}w)}{(\frac{475}{6}w)}=\)

\((\frac{75}{2}w*\frac{6}{475}w)=\)

\(\frac{225w}{475w}\)

Divide by 25:

X finishes \(\frac{9}{19}\) of productionAnswer E
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