Fun math puzzles, I'll try my best to do them weekly or something like that.

1. C=59(F−32)

The equation above shows how temperature F, measured in degrees Fahrenheit, relates to a temperature C, measured in degrees Celsius. Based on the equation, which of the following must be true?

I. A temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of 59 degree Celsius.

II. A temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit.

III. A temperature increase of 59 degree Fahrenheit is equivalent to a temperature increase of 1 degree Celsius.

A) I only

B) II only

C) III only

D) I and II only

2. The equation 24x2+25x−47ax−2=−8x−3−53ax−2 is true for all values of x≠2a, where a is a constant.

What is the value of a?

A) -16

B) -3

C) 3

D) 16

3. If 3x−y=12, what is the value of 8x2y?

A) 212

B) 44

C) 82

D) The value cannot be determined from the information given.

4. Points A and B lie on a circle with radius 1, and arc AB has a length of 3\(\pi\). What fraction of the circumference of the circle is the length of arc AB?

(I don't know how to make latex for arcs)

I'll post the answers and an explanation for each problem when every problem has an attempt.

AUnVerifiedTaxPayer Oct 6, 2024

#2**+2 **

As an input: On question 1 ..... everywhere you have the number '59' that should be ' 5/9 '

ElectricPavlov Oct 6, 2024

#3**+3 **

ummm...I would give a solid solution...but this was answered VERY well here.

1. https://www.doubtnut.com/qna/181168330

answer is d, btw :)

NotThatSmart Oct 7, 2024

#4**+3 **

2. I don't have much time to explain...hopefully you understand my work...sorry

First, put everything on the right side into one fraction

\(\frac{24x^{2}+25x-47}{ax-2}=\frac{(-8x-3)(ax-2)-53}{ax-2}\)

Simplify the numerator on right side.

\(\frac{24x^{2}+25x-47}{ax-2}=\frac{-8ax^{2}+16x-3ax+6-53}{ax-2}\) \(\frac{24x^{2}+25x-47}{ax-2}=\frac{-8ax^{2}+(-3a+16)x-47}{ax-2}\)

Since both have the same denominator, their numerators must be equal.

\(24x^{2}+25x-47=-8ax^{2}+(-3a+16)x-47\)

Since the coefficients must correspond, we can right the equation

\(24x^{2}=-8ax^{2}\) \(24=-8a\) \(a=-3\)

Thus, b is the right answer.

Thanks! :)

NotThatSmart Oct 7, 2024

#7**+2 **

Heres a solution that I found.

Multiply each side of the given equation by ax−2 (so you can get rid of the fraction). When you multiply each side by ax−2, you should have:

24x^2+25x−47=(−8x−3)(ax−2)−53

You should then multiply (−8x−3) and (ax−2) using FOIL.

24x^2+25x−47=−8ax^2−3ax+16x+6−53

Then, reduce on the right side of the equation

24x^2+25x−47=−8ax^2−3ax+16x−47

Since the coefficients of the x^2 term have to be equal on both sides of the equation, −8a=24, or a=−3.

The final answer is B.

I'm sorry for the exponents and the fractions, I'm just new to latex and don't understand it.

AUnVerifiedTaxPayer
Oct 9, 2024

#5**+3 **

3. I'm assuming we are trying to find the value of \(\frac{8^{x}}{2^{y}}\)

Let's cite some equations that could help us in these equations. There are two main ideas we will be applying for this problem.

\(a^{bc}=(a^{b})^{c}\) \(\frac{a^{b}}{a^{c}}=a^{b-c}\)

Now, let's focus on what we are trying to find. Let's simplify it a bit to get a better shot at figuring out our final answer.

\(\frac{8^{x}}{2^{y}}=\frac{(2^{3})^{x}}{2^{y}}\)

This is where we apply the second equation stated. This allows us to put x and y into one exponent.

\(\frac{2^{3x}}{2^{y}}=2^{3x-y}\)

Wait! We recognize this 3x - y from somewhere! The value was given to us in the problem! It was stated that \(3x-y=12\)

Thus, we plug in this value, we have

\(2^{12} = 4096\)

Thus, our final answer is 4096.

Thanks! :)

NotThatSmart Oct 7, 2024

#8**+2 **

Sorry, the answer choices got mixed up as well, here's the original problem.

If 3x−y=12, what is the value of 8x/2y?

A) 2^12

B) 4^4

C) 8^2

D) The value cannot be determined from the information given.

here's my solution.

One way is to write

8x/2y

so that the numerator and denominator are writen with the same base. Since 2 and 8 are powers of 2, pluging in 2^3 for 8 in the numerator of 8x/2y gives

(2^3)x/2y

which can be written

2^3*x/2y (confusing)

Since the numerator and denominator of have a common base, this expression can be written as 2(3x−y). In the problem, it says that 3x−y=12, so can substitute 12 for the exponent, 3x−y, which means that

8x/2y=2^12

AUnVerifiedTaxPayer
Oct 9, 2024