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

Question

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

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

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cdrectenwald · Answer

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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Purnank · Answer

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.

tgsankar10 · Answer

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{	ext{LCM of numerators}}{	ext{HCF of denominators}}=\frac{9*7*5*4}{4}=315

bumpbot · Answer

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

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