Four logs of woods of lengths 5 1/4 m, 1 13/15 m, 3 1/2 m and 4 9/10 m

can you explain why you did the highlighted step please?

Explanation:

Method Used: HCF & LCM from Number System Theory

= HCF (21/4, 28/15, 7/2, 49/10)
= HCF(21,28,7,49)/LCM(4,15,2,10)
=7/60

Total number of pieces: 21/4 * 60/7 + 28/15 * 60/7 + 7/2 * 60/7 + 49/10 * 60/7
= 45 + 16 + 30 + 42 = 133

IMO-C

Let:
\(a=5\frac{1}{4}\)
\(b=1\frac{13}{15}\)
\(c=3\frac{1}{2}\)
\(d=4\frac{9}{10}\)

The LCM of the four denominators is 60.
Multiplying each equation by 60, we get:

\(60a=60(5+\frac{1}{4})\)
\(60a=300+15\)
\(60a=315\)
\(60a=3*3*5*7\)

\(60b=60(1+\frac{13}{15})\)
\(60b=60+52\)
\(60b=112\)
\(60b=2*2*2*2*7\)

\(60c=60(3+\frac{1}{2})\)
\(60c=180+30\)
\(60c=210\)
\(60c=2*3*5*7\)

\(60d=60(4+\frac{9}{10})\)
\(60d=240+54\)
\(60d=294\)
\(60d=2*3*7*7\)

In the resulting blue equations, the GCF on the right side is 7.
Divide each equation by 60, applying 60 on the right side to the GCF of 7:

\(60a=3*3*5*7\)
\(a = (3*3*5)(\frac{7}{60}) = 45*\frac{7}{60}\)
Implication:
\(a\) can be split into 45 smaller logs, each of length \(\frac{7}{60}\)

\(60b=2*2*2*2*7\)
\(b = (2*2*2*2)(\frac{7}{60}) = 16*\frac{7}{60}\)
Implication:
\(b\) can be split into 16 smaller logs, each of length \(\frac{7}{60}\)

\(60c=2*3*5*7\)
\(c = (2*3*5)(\frac{7}{60}) = 30*\frac{7}{60}\)
Implication:
\(c\) can be split into 30 smaller logs, each of length \(\frac{7}{60}\)

\(60d=2*3*7*7\)
\(d = (2*3*7)(\frac{7}{60}) =42*\frac{7}{60}\)
Implication:
\(d\) can be split into 42 smaller logs, each of length \(\frac{7}{60}\)

Total number of smaller logs \(= 45+16+30+42=133\)

Hi,

I got it that we need to take GCD of numerator and LCM of denominator to solve the question, but what's the logic behind doing that? Can someone please explain as I am not able to understand the logic of this solution?

Thanks !

we need to cut these four logs in equal length that makes minimum number of pieces.
And GCD divides the given lengths in maximum equal parts.
suppose I have three pieces of length 4,8 and 16.
I can divide these of equal length of 2 to produce (2 + 4 + 8) 14 pieces
but when I divide these by 4, GCD in this case, it produces (1+ 2 + 4) 7 pieces. this is minimum number of pieces of equal length that we can produce from these logs.

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