Question

A 2-column table with 5 rows. Column 1 is labeled Days to Set Up with entries 1.5, 3, 4.5, x, 7.5. Column 2 is labeled Number of Concerts with entries 1, 2, 3, 4, 5.
Stage hands set up a new stage for a concert in the arena every 1.5 days.

Use proportional reasoning to find the value of x that completes the table showing this relationship.

Answer 1

This directly proportional relationship between p and q is written as p∝q  where that middle sign is the sign of proportionality. The value of x is 6.

What are directly proportional and inversely proportional relationships?

Let there are two variables p and q

Then, p and q are said to be directly proportional to each other if

 p = kq

where k is some constant number called the constant of proportionality.

This directly proportional relationship between p and q is written as

 p ∝ q  where that middle sign is the sign of proportionality.

In a directly proportional relationship, increasing one variable will increase another.

Now let m and n be two variables.

Then m and n are said to be inversely proportional to each other if

$m = \dfrac{c}{n} \\\\ \text{or} \\\\ n = \dfrac{c}{m}$

(both are equal)

where c is a constant number called the constant of proportionality.

This inversely proportional relationship is denoted by

$m \propto \dfrac{1}{n} \\\\ \text{or} \\\\n \propto \dfrac{1}{m}$

As visible, increasing one variable will decrease the other variable if both are inversely proportional.

Let the days to set up with entries be represented by a, while the number of concerts is represented by b. Now, It is known that the days to set up with entries and the number of the concert are in a relationship. Therefore, we can write the relationship as,

a ∝ b

a = k b

Now, if we substitute the values from the first row, we will get,

1.5 = k × 1

1.5 = k

Thus, the value of k is 1.5.

Now, if we substitute the entries from the 4th row, we will get,

a = k × b

x = 1.5 × 4

x = 6

Hence, the value of x is 6.

Learn more about Directly and Inversely proportional relationships:

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