This is the second part of the full solution series for the MTAP 2015 Questions with solutions. You may also want to read the solutions for problems **1 – 10** and **download** the questions.

**11.)** Danny had some stickers. He gave 1/3 of the stickers plus 1 sticker to his brother. Then, he gave 1/3 of the remaining stickers plus 4 stickers to his sister. Finally, he gave ½ of what remained plus 3 stickers to best friend. He found that he had 4 stickers left. How many stickers did Danny have at first?

**Solution**

If we let = number of stickers

= number of stickers given to his brothers. If we subtract the stickers given to his brother, we have

(*)

which is the number of remaining stickers.

Now, Danny gave 1/3 of this remaining stickers plus 4 stickers to his sister. That is, he gave

(#) stickers.

Now, subtracting the stickers given to her sister (#) from the remaining stickers (*), we have

(**).

Next, he gave 1/2 of the remaining stickers (**) and an additional 3. That is, he gave

(##).

Now, subtracting ## from **, we have

..

After this, only 4 stickers are left.

So, giving us .

* Answer*: 42

**12.)** What number is 2/5 of the way from – 3 to 5?

*Solution*

The distance between -3 and 5 is 8. So, (2/5)(8) = 16/5.

Now,

** Answer:** 1/5

**13.)** The number is divisible by 3. The number is . What digit does represent?

*Solution*

Since the number is divisible by 3, the number is divisible by 9. By the divisibility rules, a number is divisible by 9 if the sum of its digits is divisible by 9. Therefore, must be divisible by . Now, since is a 1-digit number, the only possible value for since and 27 is divisible by 9. Therefore, .

* Answer:* 8

**14.)** What is the smallest positive integer that must be multiplied to 60 to get a perfect cube?

**Solution**

The prime factorization of 60 is . The smallest possible number that we are looking for is .

Therefore,

.

**Answer: 450**

**15.)** Compute

**Solution**

Simplifying, we have

Now,

or .

**Answer:**

**16.)** What three-digit number is both a square of an integer and a cube of an integer?

**Solution**

The only numbers that will produce a 3-digit number when cubed are the numbers from 5 to 9. The only perfect square among this numbers is 9. Therefore, the answer is . Note that its square root is 27.

* Answer*: 729

**17.)** Simplify

*Solution*

* Answer:* where (from the restriction in the original expression)

**18.)** If x is twice as far as -9 as it is from 3, what are the possible values of x?

Solution

The distance between -9 and 3 is 12. Now 2/3 (12) = 8 and -9 + 8 = -1. Therefore, the distance from -9 to 1 is twice that distance from -1 to 3.

Also, if we let 3 be the center of the circle, we can draw a diameter from -9 to 15. Therefore, x = 15 is twice the length from -9 than from 3.

Answer: x = 1, x = 15

**19.)** Sandra is 18 years older than Paulo. In 13 years, Sandra will be as twice as old as Paulo will be then. How old is Sandra now?

**Solution**

Let x = age of Paulo

x + 18 = age of Sandra

In 13 years, Sandra will be twice as old as Paulo. That is,

2(x + 13) = x + 13 + 18

Simplifying, we have

2x + 26 = x + 31

x = 5

So, Sandra is 5 + 18 = 23 years old now.

* Answer: *23

**20.)** Which is bigger or ?

**Solution **

and .

Converting the exponents to similar fractions, we have

and .

Converting them back to radicals, we have

.

.

Since , .

**Answer:**