Grade 7 MTAP 2015 Questions with Solutions Part 2

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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.

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,

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

Answer: 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.

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 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.

Answer: 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.

Answer: 729

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.

Answer: 23

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

Solution

\sqrt[3]{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[6]{5^2} = \sqrt[6]{25}.

3^{\frac{3}{6}} = \sqrt[6]{3^3} = \sqrt[6]{27}.

Since \sqrt[6]{27} > \sqrt[6]{25}, \sqrt{3} > \sqrt[3]{5}.

Answer: \sqrt{3}

11 Comments




  3. Number 19,

    On the equation, “2x+26 = x+33” the 33 there should be 31. Because on “2(x+13) = x+13+18”, 13+18= 31. Thnkx

    Reply



  6. nice, so interesting and helpful for me. coz im the one of contestants in mtap

    Reply

  8. At #18,
    Using |x+9|=2|x-3| I think is a better way to solve the problem.

    Reply

  9. the other number is to hard to understand “for me”

    Reply

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