This is the third part (questions 21-30) of the solutions of the Grade 8 MTAP 2015 Elimination Questions. You can read the solutions for questions **1-10** and **2-10**.

Although much care was given in solving these problems, the solver is only human. If you found any errors in the solution, please comment in the box below.

**Note:** If you have old MTAP questions, you can send them to mtapreviewers@gmail.com and I will solve them for you and post it here.

**21.)** When is divided by , the quotient and remainder are both . What is ?

**Solution**

If we divide 5 by 3, then we get a quotient of 1 and a remainder of 2. If we generalize latex the dividend, as the divisor, as the quotient and as the remainder, we can form the equation

(*).

In the given, we can see that , , and .

Substituting in *,

Now .

**Answer:** -4

**22.)** If two more than twice *p* is four less than twice *q*, express *q* in terms of *p*.

**Solution**

**Answer:**

**23.)** If* x* is nonnegative and 3*x* – 4 ≤ *x*, what is the least value of *x*?

**Solution**

3*x* – 4 ≤ *x*

3*x* – *x* ≤ 4

2*x* ≤ 4

*x* ≤ 2

So the lowest non-negative value is *x* = 0.

**Answer:** *x* = 0

**24.)** In Item 23, what is the maximum value of *x*?

Answer: *x* = 2 (Obvious!)

**25.)** Solve for in the equation .

**Solution**

**Answer:** 56/55

**26.)** Solve for in the equation .

There are two values for :

Also,

.

**Answer:** -1 and 4

**27.)** If ∠B is the complement of ∠A, and the supplement of ∠A is 138°, what is ∠B – ∠A?

**Solution**

Complementary angles add up to 90° and supplementary angles add up to 180°. So,

∠A + 138 = 180

∠A = 42.

Now since ∠A and ∠B are complementary,

∠A + ∠B = 90

42 + ∠B = 90

∠B = 90 – 42

∠B = 48.

Now, ∠B – ∠A = 48 – 42 = 6

**Answer:** 6°

**28.)** If the diagonals of rectangle ABCD meet E and ∠AEB = 140°, what is ∠EBC?

**Solution**

∠A + ∠B + ∠E = 180°

∠A + ∠B + 140 = 180°

∠A + ∠B = 40°

Since ABE is an isosceles triangle, ∠A = ∠B. This means that ∠A = ∠B = 20°.

Clearly, ∠ABC = 90°. So,

∠ABE + ∠EBC = 90°

20° + ∠EBC = 90°

∠EBC = 90 – 20°

∠EBC = 70°

**Answer:** 70°

**29.)** If two exterior angles of a triangle measures 80° and 130°, what is its smallest interior angle?

**Solution**

The adjacent interior and exterior angles of a triangle are supplementary. Therefore, the largest exterior angle has the smallest adjacent interior angle.

The sum of the exterior angle of any polygon is 360°so,

Among the three angles, 150° is the largest and its adjacent interior angle is 30°.

**Answer:** 30°

**30.)** In a grouped data, if 100 – 200 forms one class, what is the class interval of the data?

Answer:

**Solution**

The class interval is the difference between the upper class limit and the lower class limit of a class. Here, the upper class limit is 200 and the lower class limit is 100. So, the class interval is 200 – 100 = 100.

**Answer:** 100