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What is the sum of the multiples of 7 from 84 to 140?

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by Brent@GMATPrepNow » Mon Sep 25, 2017 6:17 pm
What is the sum of the multiples of 7 from 84 to 140, inclusive?

A)896
B)963
c)1008
D)1792
E)2016
In other words, 84 + 91 + 98 + . . . 140 = ?

Since the values are EQUALLY SPACED, we can use the rule: SUM = [(FIRST + LAST)/2][# of values]

NUMBER of values
Here's a nice rule: If x and y are multiples of k, then the number of multiples of k from x to y inclusive = [(y-x)/k] + 1
So, for example, the NUMBER multiples of 3 from 6 to 21 inclusive = [(21 - 6)/3] + 1 = [15/3] + 1 = 6

So, the NUMBER multiples of 7 from 84 to 140 inclusive = [(140 - 84)/7] + 1
= [56/7] + 1
= 9

------------------------------------

Now apply the formula:
SUM = [(FIRST + LAST)/2][# of values]
= [(84 + 140)/2][9]
= [224/2][9]
= [112][9]
= 1008
= C

Cheers,
Brent
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by Matt@VeritasPrep » Tue Sep 26, 2017 5:16 pm
We want 7*12 + 7*13 + ... * 7*20

To find this, we could add the first 20 multiples of 7, then subtract the ones we don't want (the first 11 multiples of 7).

That gives us:

(7*1 + ... + 7*20) - (7*1 + ... + 7*11)

From there, we could write this as:

7*(1 + 2 + ... + 20) - 7*(1 + 2 + ... + 11)

The two sums in parentheses are just triangular numbers, and the summation is a cinch. :)
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by pannalal » Tue Sep 26, 2017 9:48 pm
Vincen wrote:What is the sum of the multiples of 7 from 84 to 140, inclusive?

A) 896
B) 963
C) 1008
D) 1792
E) 2016

The OA is C.

Is there any way to solve it without listing the numbers?
Sum of the multiples of 7 from 84 to 140, inclusive = 84 + 91 + ... + 140.

Number of multiples of 7 from 84 to 140 = 140/7 - 84/7 + 1 = 20 - 12 + 1 = 9.

Now, write these numbers twice, once in ascending order and then in descending order:

S = 84 + 91 + .... + 133 + 140
S = 140 + 133 + .... + 91 + 84

The sum of each pair is 224. As the total number of pairs are 9.

2S = 224*9 or S = 112*9 = 1008.

Thus, the correct answer is C.
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What is the sum of the multiples of 7 from 84 to 140?

by EconomistGMATTutor » Fri Oct 13, 2017 10:28 am
What is the sum of the multiples of 7 from 84 to 140, inclusive?


Hi Vincen,
Let's take a look at your question.

First of all we need to find the number of terms between 84 to 140 inclusive.
Since we are talking about multiples of 7, lets find out the term numbers of 84 and 140 in the sequene of multiples of 7.
$$\frac{84}{7}=12$$
84 is 12th term.

$$\frac{140}{7}=20$$
140 is 20th term.

Total number of terms = n = 20 - 12 + 1 = 9

Sum of n terms in geometric series$$ =\frac{n}{2}\left(a_1+a_n\right)$$
$$=\frac{9}{2}\left(84+140\right)$$
$$=\frac{9}{2}\left(224\right)$$
$$=\frac{9}{2}\left(84+140\right)$$
$$=9\left(112\right)$$
$$=1008$$

Therefore, Option C is correct.
Hope this helps.

I am available, if you'd like any follow up.
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by Scott@TargetTestPrep » Fri Dec 15, 2017 10:23 am
Vincen wrote:What is the sum of the multiples of 7 from 84 to 140, inclusive?

A) 896
B) 963
C) 1008
D) 1792
E) 2016
We can use the following equation to determine the quantity:

(largest multiple in the set - smallest multiple in the set)/7 + 1 = number of multiples of 7

(140 - 84)/7 + 1

56/7 + 1 = 9

Next, let's determine the average of this evenly spaced set.

average = (first number + last number)/2

average = (84 + 140)/2 = 224/2 = 112

Thus, the sum is 9 x 112 = 1,008.

Answer: C

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by Brent@GMATPrepNow » Fri Dec 15, 2017 1:26 pm
Vincen wrote:What is the sum of the multiples of 7 from 84 to 140, inclusive?

A) 896
B) 963
C) 1008
D) 1792
E) 2016

The OA is C.

Is there any way to solve it without listing the numbers?
In other words, 84 + 91 + 98 + . . . 140 = ?

Since the values are EQUALLY SPACED, we can use the rule: SUM = [(FIRST + LAST)/2][# of values]

NUMBER of values
Here's a nice rule: If x and y are multiples of k, then the number of multiples of k from x to y inclusive = [(y-x)/k] + 1
So, for example, the NUMBER multiples of 3 from 6 to 21 inclusive = [(21 - 6)/3] + 1 = [15/3] + 1 = 6

So, the NUMBER multiples of 7 from 84 to 140 inclusive = [(140 - 84)/7] + 1
= [56/7] + 1
= 9

------------------------------------

Now apply the formula:
SUM = [(FIRST + LAST)/2][# of values]
= [(84 + 140)/2][9]
= [224/2][9]
= [112][9]
= 1008
= C

Cheers,
Brent
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by Scott@TargetTestPrep » Mon Sep 23, 2019 4:37 pm
Vincen wrote:What is the sum of the multiples of 7 from 84 to 140, inclusive?

A) 896
B) 963
C) 1008
D) 1792
E) 2016

The OA is C.

Is there any way to solve it without listing the numbers?

We can use the following formula:

sum = average x quantity

Since we have an evenly spaced set of integers, we can calculate the average of the set by using this formula:

average = (smallest integer in set + greatest integer in set)/2

average = (84 + 140)/2 = 224/2 = 112

We can determine the number of multiples of 7 in the interval by using the formula:

quantity = (greatest multiple of 7 - least multiple of 7) / 7 + 1

quantity = (140 - 84)/7 + 1 = 56/7 + 1 = 9

Thus:

sum = 112 x 9 = 1008

Answer: C

Scott Woodbury-Stewart
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