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The four members of a jazz quartet all play in different rhythms. The

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Bunuel
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The four members of a jazz quartet all play in different rhythms. The [#permalink] The four members of a jazz quartet all play in different rhythms. The   07 Aug 2024, 01:30
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­The four members of a jazz quartet all play in different rhythms. The pianist plays in 9/8 time, meaning that his downbeats occur every 9/8 of a measure of time. Meanwhile, the saxophonist plays in 7/4 time, the harpist in 5/8 time, and the drummer in plain old 4/4 time (all with respect to the same measure of time as the pianist). If all four musicians start a song together on the same downbeat, how many measures later will all their downbeats occur simultaneously?

(A) 315
(B) 630
(C) 1,260
(D) 2,520
(E) 5,040

­
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A
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cdrectenwald
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Re: The four members of a jazz quartet all play in different rhythms. The [#permalink] The four members of a jazz quartet all play in different rhythms. The   07 Aug 2024, 13:11
To determine when the four musicians' downbeats will align, we need to find the least common multiple (LCM) of their rhythm intervals in terms of measures.

Intervals in Measures:

Pianist:
9
8
8
9
​
measures
Saxophonist:
7
4
=
14
8
4
7
​
=
8
14
​
measures
Harpist:
5
8
8
5
​
measures
Drummer:
1
=
8
8
1=
8
8
​
measure
Find the LCM of the Numerators
9
,
14
,
5
,
9,14,5, and
8
8:

Prime factorizations:
9
=
3
2
9=3
2

14
=
2
×
7
14=2×7
5
=
5
5=5
8
=
2
3
8=2
3

LCM includes each prime to the highest power:
LCM
=
2
3
×
3
2
×
5
×
7
=
8
×
9
×
5
×
7
=
2520
LCM=2
3
×3
2
×5×7=8×9×5×7=2520
Divide by the Common Denominator (8):

2520
8
=
315
8
2520
​
=315
Therefore, all four musicians' downbeats will align every 315 measures.

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The four members of a jazz quartet all play in different rhythms. The [#permalink] The four members of a jazz quartet all play in different rhythms. The   Updated on: 08 Aug 2024, 02:11
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Bunuel
­The four members of a jazz quartet all play in different rhythms. The pianist plays in 9/8 time, meaning that his downbeats occur every 9/8 of a measure of time. Meanwhile, the saxophonist plays in 7/4 time, the harpist in 5/8 time, and the drummer in plain old 4/4 time (all with respect to the same measure of time as the pianist). If all four musicians start a song together on the same downbeat, how many measures later will all their downbeats occur simultaneously?
(A) 315
(B) 630
(C) 1,260
(D) 2,520
(E) 5,040
­
­The pianist plays in 9/8 time, saxophonist plays in 7/4 time, the harpist in 5/8 time, and the drummer in plain old 4/4 time
Means LCM of Pianist, Saxophone and Harpist will be the one occurance when all their downbeats occur simultaneously.
\(\frac{LCMofNumerator}{HCFofDinominator} = \frac{9*7*5*4}{4} = 315\)
Answer is A.­

Originally posted by Purnank on 07 Aug 2024, 21:12.
Last edited by Purnank on 08 Aug 2024, 02:11, edited 2 times in total.
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The four members of a jazz quartet all play in different rhythms. The [#permalink] The four members of a jazz quartet all play in different rhythms. The   07 Aug 2024, 23:41
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Pianist - \(\frac{9}{8}\)
Saxophonist - \(\frac{7}{4}\)
Harpist - \(\frac{5}{8}\)
Drummer - \(\frac{4}{4}\)

Since these numbers are in fraction, lets convert it to integers. To do that, multiply each by 8. We will get: \(9, 14, 5, 8\)

This measns that in 8 time measures the Pianist played 9 downbeats, Saxophonist played 14 and so on

All will sync at their LCM:

\(LCM = 9*2*7*5*8=9*7*5*8\)

They will sync 8 times in \(9*7*5*8\) time measures. So the first time they sync will be after \(9*7*5=315\) time measures­

Answer is A

Alternative Approach:

\(\frac{\text{LCM of numerators}}{\text{HCF of denominators­}}=\frac{9*7*5*4}{4}=315\)­
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The four members of a jazz quartet all play in different rhythms. The [#permalink]
07 Aug 2024, 23:41
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Bunuel
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