# 2015 MTAP reviewer for 4th year solution part 1

This is the first part of solution series for 2015 4th year Metrobank MTAP Dep-Ed Mathematics Challenge (MMC) elimination paper.

Disclaimer: This is an author’s solution and not the official solution presented by MTAP/ Math teachers. This blog is not affiliated with MTAP or Metrobank in any way.

Problem 1:

An item was already discounted by 10% but had to be discounted by another 10% to make the price even more attractive to customers. Overall, by how many percent was the item discounted?

Solution:

Let x be the original price of the item. If 10% was taken off, the discounted price must be x-0.1x or 0.9x. If another 10% was taken off, the remaining price must be 0.9x-0.1(0.9x)=.81x. This is 0.29x less than the original price. Thus, the total discount is 29%.

Problem 2:

If the number x-4, 4-x, and x form an arithmetic progression, what is x?

Solution:

If an odd number of terms form an arithmetic progression, the median number(middle number if the terms are arrange in increasing/decreasing order) is the average of all numbers. Using this concept, 4-x is the average of x and x-4. $4-x=\dfrac{x+x-4}{2}$ $2(4-x)=2x-4$ $8-2x=2x-4$ $-2x-2x=-4-8$ $-4x=-12$ $x=3$

Problem 3:

Two sides of a triangle have lengths 15 and 25. If the third side is also a whole number, what is its shortest possible length?

Solution:

To answer this question, we take range of the third side by using the triangle inequality theorem. Third side must lie between the difference and sum of the given sides. Thus, the third side lies between 25-15 to 25+15 or between 10 and 40. This means that the shortest possible integral side of the triangle is 11.

Check this tutorial for more about triangle inequality theorem.

Problem 4:

Find the equation of the line that passes through (5,4) and parallel to 3x+y=1.

Solution:

To answer this question, we recall that the slope of parallel lines is the same. Thus, we find the slope of the given line and use it for the second line.

The slope of the line 3x+y=1 can be found by converting it to slope-intercept for (y=mx+b). We rewrite this equation as y=-3x+1. Thus, the slope is -3.

Using the point-slope form, we have $y-y_1=m(x-x_1)$ $y-4=-3(x-5)$ $y-4=-3x+15$ $3x+y-19=0$

Problem 5:

What is the area of triangle with sides 10, 10, 12

Solution:

To answer this problem, we draw the given figure. Given the 2 sides 6 and 10 half of the triangle, we can easily say that the third side is 8 which is the altitude of the triangle.

Hence we can find the area by finding the half of the product of base(12) and height(8). $A=\dfrac{1}{2}Bh$ $A=\dfrac{1}{2}\cdot 12\cdot 8$ $A=\dfrac{1}{2}\cdot 96$ $A=48 sq.units$