# 2015 MTAP Reviewer for Grade 9(26-30)

Problem 26:

If 4 men can paint a house in 5 days, in how many days can 10 men paint the same house?

Solution:

This problem is an application of variation. As the number of workers increases, the number of days to complete the job decreases. Thus, they are inversely proportional.

Let m be the number of people, d be the number of days, and k be the proportionality constant.

From this, we can say that $d=\dfrac{k}{m}$

Or we can rearrange this as follows, $k=md$

In variation, the proportionality constant is always constant. Thus, $10\cdot d=4\cdot 5$ $10d=20$ $d=2$

Problem 27:

If y is proportional to the cube of y and x is proportional to the fourth power of z, then y is proportional to which power of z?

Solution:

This problem is vague. y is proportional to cube of y doesn’t make sense. We will verify this problem first before writing the solution.

Problem 28:

Running at uniform speed in a race, Alice can beat Ben by 20 m, Ben can beat Carlo by 10 m and Alice can beat Carlo by 28 m. How long is the race?

Solution:

Let d be the distance of the race,

Let r1 be the rate of Alice

Let r2 be the rate of ben

Let r3 be the rate of Carlo

Formulating equation:

For Alice:

d=r1(t)

For Ben:

d-20=r2(t)

Since alice finished 20 m ahead of Ben and the time is equal. Equating time we have, $\dfrac{d}{r1}=\dfrac{d-20}{r2}$   #

For “Ben can beat Carlo by 10 m”. For Ben’s rate equation.

d=r2t

For Carlo’s rate equation,

d-10=r3t

Equating the time, $\dfrac{d}{r2}=\dfrac{d-10}{r3}$

We can rewrite this as follows, $\dfrac{r3}{r2}=\dfrac{d-10}{d}$   ##

For “Alice can beat Carlo by 28 m”

Alice’s rate equation:

d=r1t

For Carlo’s equation:

d-28=r3t

equating time, $\dfrac{d}{r1}=\dfrac{d-28}{r3}$    ###

Now, we observe these 3 equations. We equate the first equation and last equation since the ratio of the distance and the rate of Alice is just the same. $\dfrac{d-20}{r2}=\dfrac{d-28}{r3}$

We can rewrite this equation as follows, $\dfrac{r3}{r2}=\dfrac{d-28}{d-20}$

But we have the ratio of Carlo’s rate to Bob’s rate in ##,

thus, $\dfrac{d-10}{d}=\dfrac{d-28}{d-20}$

Voila! we have an equation in terms of distance of the race. By cross multiplication, $(d-10)(d-20)=d(d-28)$ $d^2-30d+200=d^2-28d$ $200=30d-28d$ $2d=200$ $d=100$

Therefore, the distance of the race is 100 m.

Problem 29:

Find the measure of the vertex angle of an isosceles triangle whose base angles measure 65◦.

Solution:

An isosceles triangle has two base angles and one vertex angle.

Let x be the measure base angle.

x+65+65=180

x+130=180

x=180-130

x=50

Problem 30:

Find x if the angles of a quadrilateral measure x ◦, (2x+10)◦, (3x+20)◦ and (4x−30)◦.

Solution:

Since the sum of interior angles of quadrilateral is 360 degrees we have,

x+2x+10+3x+20+4x-30=360

10x=360

x=360/10

x=36