Below is the first part of the Grade 8 MTAP 2015 Elimination Questions with Solutions and answers. If you find any errors, please comment on the box below.

1.) Find the average of the numbers -1, 3/2, and 1/2.

**Solution**

(- 1 + 3/2 + 1/2)/3 = 1/3

**Answer:** 1/3

2.) How much larger is 2/3 than 1/6?

**Solution**

2/3 – 1/6 = 4/6 – 1/6 = 3/6 = 1/2

**Answer:** 1/2

3.) If one ream contains 500 sheets of paper and a sheet of paper is 0.3 mm thick, how thick is one ream in meters?

**Solution**

500 × 0.3mm = 150 mm = 0.15m

**Answer:** 0.15m

4.) What is the second largest number among numbers , 3/2 , 1.4, and 1.6?

**Solution**

is around 1.41 and is around 1.7.

Answer: 1.6

5.) If an inch is about 2.54 cm, what is 1 cm to the nearest hundredth of an inch?

**Solution**

**Answer:** 0.39

6.) If *U *= {1, a, 2, b, 3, c, 4, d} and A = {1, 2, c, d}, what is A^{c }?

**Solution**

A^{c} is the complement of A, or the elements of the set that is not in A but in U. So,

A^{c} = {a, b, 3, 4 }

**Answer:** A^{c} = {a, b, 3, 4 }

7.) Using the same sets in Item 6 and B = {1, 2, 3, 4}, how many subsets does A ∩ B^{c} have?

**Solution**

A = {1, 2, c, d} and B = {1, 2, 3, 4}. The complement of B denoted by B^{c} are the elements of U not in B. So, B^{c} = {a, b, c, d}. Now, A ∩ B^{c} are the elements that are common to A and B^{c}. Therefore, A ∩ B^{c} = {c, d}. Now, the number of subsets of a set with n elements is 2^{n} (this includes the empty set), so there are 2^{2} = 4 subsets.

**Answer:** 4

8.) If |P| = 10, |Q| = 12, and |P Q| = 15, what is |P ∩ Q |?

**Solution 1**

We know that the cardinality of the union of two sets is equal to the sum of the cardinality of these sets less the cardinality of their intersection. That is, if we have sets P and Q, |P ∪ Q| = |P| + |Q| – |P ∩ Q|.

Substituting, we have

15 = 10 + 22 – |P ∩ Q|

|P ∩ Q| = 7

**Answer:** 7

**Solution 2**

x + y + z = 15 (*)
x + y = 12 ( )
y + z = 10 (*)

Adding the (**) and (***), we have x + 2y + z = 22 (#)

Subtracting (*) from (#),

x + 2y + z – (x + y + z) = 22 – 15

y = 7

**Answer:** 7

9.) If |M ∩ N| = 24 and |M ∪ N| = 26, what is |M| + |N|?

**Solution**

From number 8, we know that |M ∪ N| = |M| + |N| – |M ∩ N|. Substituting, we have,

26 = |M| + |N| – 24

|M| + |N| = 50.

**Answer:** 50

10.) There were 59 participants during the recent math camp. Among them, 37 liked doing projects, 30 liked solving problems, and 13 liked both. How many of the participants did not like at least one of these two activities?

**Solution **

Solution will be discussed in a separate post.

Answer: 5