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

Or we can rearrange this as follows,

In variation, the proportionality constant is always constant. Thus,

Thus, answer is 2 days.

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,

#

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,

We can rewrite this as follows,

##

For “Alice can beat Carlo by 28 m”

Alice’s rate equation:

d=r1t

For Carlo’s equation:

d-28=r3t

equating time,

###

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.

We can rewrite this equation as follows,

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

thus,

Voila! we have an equation in terms of distance of the race. By cross multiplication,

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

Answer is 50 degrees.

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