Question

an interior decorator is painting a bedroom. after painting \\(\frac{1}{10}\\) of the room, they have used \\(\frac{1}{2}\\) of a can of paint.

select all of the true statements.

\\(\square\\) the entire bedroom requires 20 cans of paint.

\\(\square\\) each can of paint will cover \\(\frac{1}{20}\\) of the bedroom.

\\(\square\\) 3 cans of paint will cover 60% of the bedroom.

\\(\square\\) the entire bedroom requires 5 cans of paint.

\\(\square\\) painting \\(\frac{1}{2}\\) of the bedroom requires 10 cans of paint.

Explanation:

Step1: Define the relationship

Let \( x \) be the fraction of the room covered by 1 can of paint. We know that painting \( \frac{1}{10} \) of the room uses \( \frac{1}{2} \) can of paint. So we set up the proportion: \( \frac{\frac{1}{10}}{\frac{1}{2}}=\frac{1}{x} \) (or we can think of it as: if \( \frac{1}{2} \) can covers \( \frac{1}{10} \) of the room, then 1 can covers \( \frac{1}{10}\div\frac{1}{2}=\frac{1}{10}\times2 = \frac{1}{5} \) of the room? Wait, no, wait. Wait, actually, let's re - express. Let \( C \) be the number of cans needed for the whole room. If \( \frac{1}{2} \) can is used for \( \frac{1}{10} \) of the room, then the ratio of the fraction of the room painted to the number of cans used is constant. So \( \frac{\frac{1}{10}}{\frac{1}{2}}=\frac{1}{C} \). Solving for \( C \): \( C=\frac{\frac{1}{2}}{\frac{1}{10}}=\frac{1}{2}\times10 = 5 \) cans for the whole room.

Step2: Analyze each statement

• Statement 1: The entire bedroom requires 20 cans of paint.

From our calculation, the whole room requires 5 cans, so this is false.

• Statement 2: Each can of paint will cover \( \frac{1}{20} \) of the bedroom.

If 5 cans cover 1 whole room, then 1 can covers \( \frac{1}{5} \) of the room, so this is false.

• Statement 3: 3 cans of paint will cover 60% of the bedroom.

1 can covers \( \frac{1}{5}=0.2 = 20\% \) of the room. So 3 cans cover \( 3\times20\% = 60\% \) of the room. This is true.

• Statement 4: The entire bedroom requires 5 cans of paint.

As calculated earlier, \( \frac{1}{2} \) can covers \( \frac{1}{10} \) of the room. So the number of cans for the whole room \( C=\frac{1}{\frac{1}{10}\div\frac{1}{2}}=\frac{1}{\frac{1}{5}} = 5 \). This is true.

• Statement 5: Painting \( \frac{1}{2} \) of the bedroom requires 10 cans of paint.

If 5 cans cover the whole room, then \( \frac{1}{2} \) of the room requires \( \frac{5}{2}=2.5 \) cans, not 10. So this is false.

Answer:

3 cans of paint will cover 60% of the bedroom.
The entire bedroom requires 5 cans of paint.