Sum/difference:
Let

This means that

Now, assume that
is rational. The sum/difference of two rational numbers is still rational (so 5-x is rational), and the division by 3 doesn't change this. So, you have that the square root of 8 equals a rational number, which is false. The mistake must have been supposing that
was rational, which proves that the sum/difference of the two given terms was irrational
Multiplication/division:
The logic is actually the same: if we multiply the two terms we get

if again we assume x to be rational, we have

But if x is rational, so is -x/15, and again we come to a contradiction: we have the square root of 8 on one side, which is irrational, and -x/15 on the other, which is rational. So, again, x must have been irrational. You can prove the same claim for the division in a totally similar fashion.
Answer:
25% is called a quarter of 900
Step-by-step explanation:

Answer:
Option A is the correct choice.
Step-by-step explanation:
Let d be the number of boxes of duck calls and t be the number of boxes of turkey calls.
We have been given that a company sells boxes of duck calls for $35 and boxes of turkey calls (t) for $45, so the revenue earned from selling d boxes of duck and t boxes of turkey call will be 35d and 45t respectively.
Further, the company plan to make $300. We can represent this information as:

We are also told that they make batches of duck calls that fill 6 boxes and batches of turkey calls that fill 8 boxes. the company only has 42 boxes. We can represent this information as:


Therefore, our desired system of equation will be:

Answer:
We conclude that the price is 3.5 times the number of board games.
Hence, option B is true.
Step-by-step explanation:
We know that when y varies directly with x, the equation is
y ∝ x
y = kx
k = y/x
where 'k' is called the proportionality constant.
From the table,
For the point (2, 7)
k = y/x
= 7/2
= 3.5
For the point (4, 14)
k = y/x
= 14 / 4
= 7/2
= 3.5
For the point (5, 17.50)
k = y/x
= 17.5 / 5
= 3.5
For the point (9, 31.50)
k = y/x
= 31.50 / 9
= 3.5
From the above calculations, we computed that the value of the proportionality constant remains the same.
Thus, the table of numbers represents a proportional relationship.
Therefore, the equation becomes
y = kx
The price of 2 board game
y = 3.5 (2)
= 7
The price of 4 board game
y = 3.5 (4)
= 14
Therefore, we conclude that the price is 3.5 times the number of board games.
Hence, option B is true.