# Grade 7 MTAP 2015 Questions with Solutions Part 2

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 $x$ = number of stickers $\frac{1}{3}x + 1$ = number of stickers given to his brothers. If we subtract the stickers given to his brother, we have $x - (\frac{1}{3}x + 1) = \frac{2}{3}x - 1$ (*)

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 $\frac{1}{3} (\frac{2}{3}x - 1) + 4 = \frac{2}{9}x + \frac{11}{3}$ (#) stickers.

Now, subtracting the stickers given to her sister (#) from the remaining stickers (*), we have $(\frac{2}{3}x - 1 - (\frac{2}{9}x + \frac{11}{3} = \frac{4}{9}x - \frac{14}{3}$ (**).

Next, he gave 1/2 of the remaining stickers (**) and an additional 3. That is, he gave $\frac{1}{2} (\frac{4}{9}x - \frac{14}{3}) = \frac{2}{9}x + \frac{2}{3}$ (##).

Now, subtracting ## from **, we have $(\frac{4}{9}x - \frac{14}{3}) - (\frac{2}{9}x - \frac{2}{3}) = \frac{2}{9}x - \frac{16}{3}$..

After this, only 4 stickers are left.

So, $\frac{2}{9}x - \frac{16}{3} = 4$ giving us $x = 42$.

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, $-3 + \dfrac{16}{5} = \dfrac{-15}{5} + \dfrac{16}{5} = \dfrac{1}{5}$

13.) The number $n$ is divisible by 3. The number $n^2$ is $3641A5$. What digit does $A$ represent?

Solution
Since the number $n$ is divisible by 3, the number $n^2$ 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, $3 + 6 + 4 + 1 + A + 5 = A + 19$ must be divisible by $9$. Now, since $A$ is a 1-digit number, the only possible value for $A = 8$ since $19 + 8 = 27$ and 27 is divisible by 9. Therefore, $A = 8$.

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 $2^2 \times 3 \times 5$. The smallest possible number that we are looking for is $2^3 \times 3^3 \times 5^3$.

Therefore, $\dfrac{2^3 \times 3^3 \times 5^3}{ 2^2 \times 3 \times 5} = 2 \times 3^2 \times 5^2 = 450$.

15.) Compute $10 \div \dfrac{1 + \frac{1}{4}}{2 - \frac{3}{5}}$

Solution

Simplifying, we have $\dfrac{1 + \frac{1}{4}}{ 2 - \frac{3}{5}} = \dfrac{\frac{5}{4}}{ \frac{7}{5}} = \dfrac{25}{28}.$

Now, $10 \div \frac{25}{28} = 10 \times \frac{28}{25} = \frac{56}{5}$ or $11 \frac{1}{5}$.

Answer: $11 \frac{1}{5}$

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 $9^3 = 729$. Note that its square root is 27.

17.) Simplify $\dfrac{(\sqrt{x} - \sqrt{3})(\sqrt{x} + \sqrt{3})}{x^2 - 3x}$

Solution $\dfrac{(\sqrt{x} - \sqrt{3})(\sqrt{x} + \sqrt{3})}{x^2 - 3x} = \dfrac{x - 3}{x(x - 3)} = \frac{1}{x}$

Answer: $\frac{1}{x}$ where $x \neq 3$ (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.

20.) Which is bigger $\sqrt{5}$ or $\sqrt{3}$?

Solution $\sqrt{5} = 5^{\frac{1}{3}}$ and $\sqrt{3} = \frac{1}{2}$.

Converting the exponents to similar fractions, we have $5^{\frac{1}{3}} = 5^ {\frac{2}{6}}$ and $3^{\frac{1}{2}}$.

Converting them back to radicals, we have $5^{\frac{2}{6}} = \sqrt{5^2} = \sqrt{25}$. $3^{\frac{3}{6}} = \sqrt{3^3} = \sqrt{27}$.

Since $\sqrt{27} > \sqrt{25}$, $\sqrt{3} > \sqrt{5}$.

Answer: $\sqrt{3}$

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