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How many multiples of 33 lie between 101 and 1,000, inclusive?

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How many multiples of 33 lie between 101 and 1,000, inclusive?  [#permalink]

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New post 20 Sep 2018, 02:04
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[Math Revolution GMAT math practice question]

How many multiples of \(33\) lie between \(101\) and \(1,000,\) inclusive?

\(A. 24\)
\(B. 27\)
\(C. 33\)
\(D. 36\)
\(E. 48\)

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Re: How many multiples of 33 lie between 101 and 1,000, inclusive?  [#permalink]

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New post 20 Sep 2018, 02:13
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MathRevolution wrote:
[Math Revolution GMAT math practice question]

How many multiples of \(33\) lie between \(101\) and \(1,000,\) inclusive?

\(A. 24\)
\(B. 27\)
\(C. 33\)
\(D. 36\)
\(E. 48\)



First multiple of 33 in the range : 132 (33*4)

Last Multiple in the range : 990 (33*30)

No. of multiple in the range : (30-4) + 1 = 27.

The best answer is B.
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Re: How many multiples of 33 lie between 101 and 1,000, inclusive?  [#permalink]

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New post 20 Sep 2018, 06:19
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MathRevolution wrote:
[Math Revolution GMAT math practice question]

How many multiples of \(33\) lie between \(101\) and \(1,000,\) inclusive?

\(A. 24\)
\(B. 27\)
\(C. 33\)
\(D. 36\)
\(E. 48\)


Some positive multiples of 33 are: 33, 66, 99, 132, 165, 198,. . . , 957, 990, 1023
So, we want the number of multiples of 33 from 132 to 990 inclusive

Observe:
132 = (33)(4)
165 = (33)(5)
198 = (33)(6)
.
.
.
957 = (33)(29)
990 = (33)(30)

We can see that the number of multiples of 33 from 132 to 990 inclusive is the SAME as the number of integers from 4 to 30 inclusive.

To determine the above, we can apply the following rule: the number of integers from x to y inclusive equals y - x + 1
So, the number of integers from 4 to 30 inclusive = 30 - 4 + 1 = 27

Answer: B

Cheers,
Brent
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Re: How many multiples of 33 lie between 101 and 1,000, inclusive?  [#permalink]

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New post 20 Sep 2018, 06:23
Top Contributor
MathRevolution wrote:
[Math Revolution GMAT math practice question]

How many multiples of \(33\) lie between \(101\) and \(1,000,\) inclusive?

\(A. 24\)
\(B. 27\)
\(C. 33\)
\(D. 36\)
\(E. 48\)


Another approach is to apply the following 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 multiples of 3 from 6 to 21 inclusive = [(21 - 6)/3] + 1 = [15/3] + 1 = 6

So, the number of multiples of 33 from 132 to 990 inclusive = (990 - 132)/33 + 1
= 858/33 + 1
= 26 + 1
= 27

Answer: B

Cheers,
Brent
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Re: How many multiples of 33 lie between 101 and 1,000, inclusive?  [#permalink]

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New post 20 Sep 2018, 06:34
MathRevolution wrote:
[Math Revolution GMAT math practice question]

How many multiples of \(33\) lie between \(101\) and \(1,000,\) inclusive?

\(A. 24\)
\(B. 27\)
\(C. 33\)
\(D. 36\)
\(E. 48\)


Since 1000 - 101 ≈ 900 -- and the answer choices are a bit spread out -- we can count the multiples of 33 simply by dividing 33 into 900:
900/33 = 300/11 = a bit more than 27.
Thus, there are 27 multiples of 33 between 101 and 1000.


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How many multiples of 33 lie between 101 and 1,000, inclusive?  [#permalink]

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New post 20 Sep 2018, 08:07
MathRevolution wrote:
[Math Revolution GMAT math practice question]

How many multiples of \(33\) lie between \(101\) and \(1,000,\) inclusive?

\(A. 24\)
\(B. 27\)
\(C. 33\)
\(D. 36\)
\(E. 48\)


One of our students´ most-loved mottos is: let the "Queen of Sciences" (Mathematics) do the weight-lifting for you!

\(101 < 33M < 1000\)

\(? = M\,\,\,\left( {\operatorname{int} } \right)\)

\(\left( {99 + 33 = } \right)\,\,\,132 \leqslant 33M \leqslant 990\,\,\,\,\left( { = 30 \cdot 33} \right)\)

\(4 \leqslant M \leqslant 30\,\,\,\, \Rightarrow \,\,\,\,? = 30 - 4 + 1 = 27\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
fskilnik.

P.S.: that´s EXACTLY Selim´s solution, only a bit more "structured". Congrats, Selim!!
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Re: How many multiples of 33 lie between 101 and 1,000, inclusive?  [#permalink]

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New post 24 Sep 2018, 06:12
=>
Consider the arithmetic sequence \(132, 165, …, 990\) of multiples of \(33\).
The number of terms in this sequence is \(\frac{(990-132)}{33} + 1 =\frac{858}{33} + 1 = 26 + 1 = 27.\)

Therefore, the answer is B.
Answer: B
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Re: How many multiples of 33 lie between 101 and 1,000, inclusive?   [#permalink] 24 Sep 2018, 06:12
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