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The sum of three integers is 40. The largest integer is 3

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The sum of three integers is 40. The largest integer is 3  [#permalink]

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New post 28 Jan 2014, 00:57
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The Official Guide For GMAT® Quantitative Review, 2ND Edition

The sum of three integers is 40. The largest integer is 3 times the middle integer, and the smallest integer is 23 less than the largest integer. What is the product of the three integers?

(A) 1,104
(B) 972
(C) 672
(D) 294
(E) 192

Problem Solving
Question: 62
Category: Algebra Simultaneous equations
Page: 69
Difficulty: 600


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Re: The sum of three integers is 40. The largest integer is 3  [#permalink]

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New post 28 Jan 2014, 00:57
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SOLUTION

The sum of three integers is 40. The largest integer is 3 times the middle integer, and the smallest integer is 23 less than the largest integer. What is the product of the three integers?

(A) 1,104
(B) 972
(C) 672
(D) 294
(E) 192

The middle number = x;
The largest integer is 3 times the middle integer = 3x;
The smallest integer is 23 less than the largest integer = 3x-23.

The sum of three integers is 40 --> x + 3x + 3x - 23 = 40 --> x=9 --> the product ((3x-23)*x*3x) is a multiple of 9. The only multiple of 9 among the options is B (9 + 7 + 2 = 18 = (a multiple of 9), which implies that 972 IS a multiple of 9).

Answer: B.
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Re: The sum of three integers is 40. The largest integer is 3  [#permalink]

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New post 28 Jan 2014, 03:29
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Set up the values as in the given question:

Let the middle value be denoted by the value 'n'. the sequence as per question is therefore:

3n-23;n;3n for the smallest, middle and largest integers respectively.
So 3n-23+n+3n=40
n=9.

After substitution, the value of the three numbers will be 4;9;27 in the same order above.

Their product =972.

Answer B.
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Re: The sum of three integers is 40. The largest integer is 3  [#permalink]

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New post 28 Jan 2014, 10:05
Cosmas wrote:
Set up the values as in the given question:

Let the middle value be denoted by the value 'n'. the sequence as per question is therefore:

3n-23;n;3n for the smallest, middle and largest integers respectively.
So 3n-23+n+3n=40
n=9.

After substitution, the value of the three numbers will be 4;9;27 in the same order above.

Their product =972.

Answer B.



I did the same way.
Agree with B...
IMO:B
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Re: The sum of three integers is 40. The largest integer is 3  [#permalink]

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New post 28 Jan 2014, 20:57
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1
Let us assume 3 different integers to be x, y and z, in ascending order.

Given that x + y + z = 40;
We are give that z = 3y & x = z - 23 (or) x = 3y - 23;(just to keep a single variable for calculation)

3y - 23 + y + 3y = 40;
7y = 63;
y = 9;

From here, we can determine that z = 27 & x = 4;

Since we are asked to find the product, we can find xyz: 4 * 9 * 27 = 972;

Ans is (B).
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Re: The sum of three integers is 40. The largest integer is 3  [#permalink]

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New post 29 Jan 2014, 02:28
1
a+b+c=40 (1)
Let a<b<c
Therefore,c=3b and a=c-23.
Replacing, a and b in (1)
we get c=27
a=4 and b=9
a*b*c=4*9*27=972
Ans. B
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Re: The sum of three integers is 40. The largest integer is 3  [#permalink]

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New post 30 Jan 2014, 10:37
When you guys post questions from the Official Review...How do you know the difficulty level?

The difficulty is crazy low, but its a 600 Level Question? :shock:


and yah, I got 972 as well

A+B+C=40

A+B+3B=40
3B-23+B+3B=40
7B=63

\(\frac{7B}{7}\) \(=\) \(\frac{63}{7}\)

\(B=9\)

The rest is history

I represent my three numbers as three different variables because it shows how they can be non or interrelated :-D when I started the problem, I had no idea about the relationship(s) of the numbers and never assume they are "n" related, besides being integers.
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Re: The sum of three integers is 40. The largest integer is 3  [#permalink]

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New post 02 Feb 2014, 06:09
The sum of three integers is 40. The largest integer is 3 times the middle integer, and the smallest integer is 23 less than the largest integer. What is the product of the three integers?

(A) 1,104
(B) 972
(C) 672
(D) 294
(E) 192


Let the numbers be \(x_1\), \(x_2\) and \(x_3\)

Given that, \(x_3 = 3*x_2\) Or, \(x_2 = x_3/3\) & \(x_1 = x_3 -23\)

Since,\(x_1+ x_2 + x_3 = 40\)

Or,\(x_3-23 +\frac{x_3}{3} + x_3=40\)

Or, \(3* x_3+x_3 + x_3*3=(40 + 23)*3\)

Or, \(7x_3=189\)

Or, \(x_3=27\)

Substituting the value of \(x_3\), \(x_1= x_3-23=4\) & \(x_2=x_3/3=9\)

So, \(x_1*x_2*x_3=4*9*27=972\)

Answer: (B)
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Re: The sum of three integers is 40. The largest integer is 3  [#permalink]

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New post 27 Jul 2015, 09:08
1
L + M + S = 40
3M + M + 3M - 23= 40
7M=63, M=9 -> L=27 -> S=4 ---> 4*27*9 ~ 100*9 = 900 (B)
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Re: The sum of three integers is 40. The largest integer is 3  [#permalink]

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New post 27 Jul 2015, 16:54
2
Hi All,

This prompt is built on a few Number Properties, so you can actually solve it with a bit of 'brute force' math (without the need of any lengthy equations or calculations:

We're given a few facts to work with:
1) The SUM of 3 integers is 40
2) The largest integer = 3(middle integer)
3) The largest integers = 23 + smallest integer

We're asked for the PRODUCT of the three integers.

First, notice how all 3 answers are POSITIVE, that means that our 3 integers are likely all positive. From the given information, we can see that the largest integer will be considerably bigger than each of the other two.

From the first fact, we know that the largest integer is 3 TIMES the middle integer, so the largest integer MUST be a multiple of 3. Here, we can start brute-forcing the work by 'playing around' with multiples of 3...

IF...
the largest number is 21, then the other two numbers are 7 and -2; the sum is 26, which is TOO LOW. Let's try something bigger....

IF....
the largest number is 24, then the other two numbers are 8 and 1; the sum is 33, which is TOO LOW. We have to go bigger...

IF....
the largest number is 27, then the other two numbers are 9 and 4; the sum IS 40, so these MUST be the 3 numbers

The answer to the question will be the product of 27, 9 and 4. You can organize this math in variety of ways, but you might find it easiest to look at it this way....

(27x4)(9) = (108)(9) --> something MORE than 900 but LESS than 1,000. You might also notice that since one of the numbers is 9, the product MUST be a multiple of 9 (and there's only one answer that fit's THAT description).

Final Answer:

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Re: The sum of three integers is 40. The largest integer is 3  [#permalink]

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New post 29 Jul 2016, 09:34
Here is my approach =>
let x,y,z be the three integers in decreasing order
here x+y+z=40
x+x/3+x-23=40
=> 7x=63*9=> x=27
so y=27/3=9
z=27-23=4
hence the product = 27*9*4
also as the only option that is a multiple of three is B => Smash it
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Re: The sum of three integers is 40. The largest integer is 3  [#permalink]

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New post 14 Aug 2017, 14:23
Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

The sum of three integers is 40. The largest integer is 3 times the middle integer, and the smallest integer is 23 less than the largest integer. What is the product of the three integers?

(A) 1,104
(B) 972
(C) 672
(D) 294
(E) 192


We can create the following 3 equations in which a = the smallest integer, b = the middle integer, and c = the largest integer:

a + b + c = 40

and

c = 3b

c/3 = b

and

c - 23 = a

Thus:

c - 23 + c/3 + c = 40

2c + c/3 = 63

Multiplying by 3, we have:

6c + c = 189

7c = 189

c = 27, so a = 27 - 23 = 4 and b = 27/3 = 9.

Thus, the product of the 3 integers is 4 x 9 x 27 = 972.

Answer: B
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Re: The sum of three integers is 40. The largest integer is 3  [#permalink]

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Re: The sum of three integers is 40. The largest integer is 3 &nbs [#permalink] 18 Oct 2018, 15:30
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