How many positive integers less than 500 begin with a 3, end with a 3,

Question

How many positive integers less than 500 begin with a 3, end with a 3, or both?

A) 149 
B) 150 
C) 159 
D) 199 
E) 200

*kudos for all correct  solutions

BrentGMATPrepNow · Accepted Answer

There are   499   integers from 1 to 499 inclusive.

Let's determine how many of those   499   integers DO NOT meet the condition of beginning with a 3, ending with a 3, or both

1-digit integers that DO NOT begin with a 3, end with a 3, or both  
1, 2, 4, 5, 6, 7, 8, and 9
TOTAL =  8

2-digit integers that DO NOT begin with a 3, end with a 3, or both  
In the tens position, we can have 1, 2, 4, 5, 6, 7, 8, or 9 ( 8  possibilities)
In the units position, we can have 0, 1, 2, 4, 5, 6, 7, 8, or 9 ( 9  possibilities)
TOTAL number of 2-digit integers that DO NOT begin with a 3, end with a 3, or both = ( 8 )( 9 ) =  72

3-digit integers that DO NOT begin with a 3, end with a 3, or both  
In the hundreds position, we can have 1, 2, or 4 ( 3  possibilities)
In the tens position, we can have 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 ( 10  possibilities)
In the units position, we can have 0, 1, 2, 4, 5, 6, 7, 8, or 9 ( 9  possibilities)
TOTAL number of 3-digit integers that DO NOT begin with a 3, end with a 3, or both = ( 3 )( 10 )( 9 ) =  270

So, the TOTAL number of integers that DO NOT meet the given condition =  8  +  72  +  270  =   350

TOTAL number of integers that MEET the given condition
So, the =   499   -   350   = 149

Answer: A

Cheers,
Brent

chetan2u · Answer

Let's see how many start with 3....
Single digit...3 -1
2-digits ...30-39 - 10
3-digits.....300-399 -100
Total 1+19+100=111

Let's see how many end with 3....
Every tens has one 3.. 3,13,23,33...,93-10
Number of tens ... 500/10=50

Numbers repeated above .. those starting and ending with 3
10 in 300s - 303,313,...393
33 in 2-digits
3 in single digit
So total 12

Ans 111+50-12=161-12=149

A

rishabhmail125 · Answer

chetan2u  Thanks a lot for the easy explanation. I just have a quick question, the no of digits between 30 and 39 that start with 3 would be 10 right? which makes the total 111? Also, there would be one more redundant value in the single digit group which is 3? Is that correct or am I missing something?

GyMrAT · Answer

Lets start with single digit numbers, we have only 1 number 3

Now 2 digit numbers we have of the form 3_ & _3
3_ are 10 numbers (30,31,32,33....39)
_3 are 9 numbers (13,23,33,.....93)
we get 33 in both sets, hence we remove 1 from total of 2 digit numbers,

Hence total 2 digit numbers is 18

Now 3 digit numbers, we have of the form 3 _ _ & _ _ 3
3 _ _ are 100 numbers (300,301,302,303,.....399)
_ _ 3 are 40 numbers (103,113,....203,213,....303,313,....413,423,...493)
we also have numbers of the form 3 _ 3 (303,313,...393) common to both sets, these are 10 such numbers.

Hence total 3 digit numbers is 100 + 40 -10 = 130

Hence the total numbers of the required form = 1 + 18 + 130 = 149.

Answer A.

Thanks,
GyM

Ps. You can also use combinatorics to count the numbers, for e.g. 3 _ _ can be 1*10C1*10C1 = 100 & _ _ 3 can be 4C1*10C1*1 = 40

US09 · Answer

The total number of 3 digit numbers that begin with 3 = 100.
The total number of numbers that end with 3 = From 0 to 100 - 10
 From 100 to 200 - 10
 From 200 to 300 - 10, Excluding all numbers ending with 3 in 300s as already counted.
 From 400 to 500 - 10
The total number of numbers beginning with 3 but are 2 digit numbers = 9 (30, 31, 32, 34, 35, 36, 37, 38, 39) - Excluding 33 as was already counted in the ending with 3 section.
Total = 100 + 40 + 9 = 149.

pushpitkc · Answer

We need to include one digit, two-digit, and three-digit numbers less than 500.

1-99 
3 is the only one digit number 
30-39 are the two digit numbers starting with 3. (Total of 10 numbers)
13,23,43,53,63,73,83,93 are the two-digit numbers ending with 3. (Total of 8 numbers)

100-199 
113,123,133,143,153,163,173,183,and 193 are the two-digit numbers ending with 3 in this range

Similarly, both the ranges  200-299  and  400-499  have 10 numbers each which end with 3.

The range  300-399  has all the hundred numbers starting with 3(we needn't include those
numbers in this range which end with 3 because that would lead to double counting)

Therefore, there are a total of  1 + 8 + 10*4 + 100 = 149  (Option A)  such numbers less than 500.

pushpitkc · Answer

Yes,  rishabhmail125  - Your observation is correct.

The numbers starting with 3 will include 33 in the list. The total numbers starting with 3 is 111. 
Also, 3 is part of both the lists. The total numbers which are part of both lists are 12 in number.

Therefore, the total numbers(starting with 3, ending with 3, or both) less than 500 are 149(111+50-12)

chetan2u · Answer

Rishabh, you are absolutely correct..
There was a typo in the post, edited now.
There will be 111 positive integers starting with 3 and 50 ending with 3 but 12 common to both so 111+50-12=149

Pritishd · Answer

Sharing my approach to this question using the slot method:

In all there are  10  digits from  0, 1, 2, 3, ... , 9  inclusive

Single Digit Number    = 1 
Only  3

Double Digit Numbers     = 18 
a. Starts and ends with 3  =  Only  33 
b. Starts with 3 but ends with a number other than 3  =   1   9   = 9 
c. Starts with a number other than 3 but ends with 3  =   8   1   = 8  (The first digit cannot be 0 and 3, hence 8)

Three Digit Numbers    = 130 
a. Starts and ends with 3  =   1   10   1 
b. Starts with 3 but ends with a number other than 3  =   1   10   9   = 90 
c. Starts with a number other than 3 but ends with 3  =   3   10   1   = 30  (Only 1, 2 and 4 can take the first place as the number has to be between 100 and 500)

Combining the results = 1 + 18 + 130 = 149 (Ans.  A )

Fdambro294 · Answer

Another fun one. Key is to avoid over counting and try to keep straight the numbers that repeat the digit 3 in the front and back (33 and 303) as over-counting either of these Numbers will land you with the wrong answer.

(1st) numbers that begin with a 3 digit

(A)One digit numbers - only 3

1 number

(B) Two digit numbers that begin with 3

Tens place: 1 available option as 3 ——-1
And
Units place: every digit except 3 ——— 9

1 * 9 = 9 Numbers

(C) Three Digit Numbers that begin with 3:

Hundreds Place: 1 available option, locked in 3 ———- 1

And

Tens Place: can take any of the 10 digits ——- 10

And

Units Place: every digit except 3 ———9

1 * 10 * 9 = 90 Numbers

So far we have: 1 + 9 + 90 = 100 Numbers.

(2nd) Numbers that End with a 3 digit (except the single digit number 3)

From 0 to 100: (note: we did not count 33 above, so we will count it here)

13, 23, 33, 43, 53, 63, 73, 83, 93 ——> 9 Numbers

103, 113, 123, ........ 193 ———-> 10 Numbers

203 .... 293 ———-> 10 Numbers

303 .....393 (note: we did not include digit 3 above when we counted numbers that began with 3) ———> 10 Numbers

403 ..... 493 ———-> 10 Numbers

9 + 10 + 10 + 10 + 10 = 49

Finally, there are - 100 + 49 =

(A) 149 Numbers

Posted from my mobile device

LaveenaPanchal · Answer

The number with hundreds digit as 3. From 300 to 399 = 100 -- (1)
The number with units digit as 3. 10 (in each hundred) x 5 = 50--(2)
Less one number taken twice i.e. 333. considered in both (1) and (2)--- (3)

The correct answer in Option A. (1) + (2) - (3) = 100 + 50 -1 = 149.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

How many positive integers less than 500 begin with a 3, end with a 3,

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

How many positive integers less than 500 begin with a 3, end with a 3,

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards