Problem 33 - Prealgebra

Question

If 1 pint of paint is needed to paint an area of 25 square feet, how many pints must be purchased in order to paint an area that is 9 feet 6 inches square?

Step-by-Step Solution

Answer

4 pints of paint are needed.

Step 1: Convert Measurements to Feet

First, convert the dimensions of the area to be painted from feet and inches to feet only. The area is given as 9 feet 6 inches. To convert 6 inches to feet, remember that there are 12 inches in a foot, so 6 inches is equivalent to \(\frac{6}{12} = 0.5\) feet. Thus, the total length is \(9 + 0.5 = 9.5\) feet.

Step 2: Calculate the Area in Square Feet

Next, calculate the area of the square using the length you found in Step 1. The side of the square is 9.5 feet, so the area is \((9.5)^2\). This results in an area of \(9.5 \times 9.5 = 90.25\) square feet.

Step 3: Determine Paint Needed

Determine the number of pints of paint required. We know that 1 pint of paint covers 25 square feet. To find out how many pints are needed for 90.25 square feet, divide 90.25 by 25. Therefore, \(\frac{90.25}{25} = 3.61\).

Step 4: Round Up to Nearest Whole Number

Since you can't purchase a fraction of a pint, and you need to cover the entire area, round 3.61 up to the nearest whole number. Thus, you will need 4 pints of paint.

Key Concepts

Understanding Area Calculation

Area calculation is an essential concept, especially when planning tasks like painting. To find the area of a square or rectangle, you multiply the length by the width. It's important to ensure that both dimensions are in the same units before performing the calculation.
For a square, both length and width are the same, so it's simply the length squared. In our exercise, the side of the square is 9.5 feet, and to calculate the area, we use the formula: \[\text{Area} = 9.5 \times 9.5\]This results in 90.25 square feet. Calculating the area helps us determine how much material is needed for coverage, like paint in this scenario. The precision in measurement directly affects the outcome, so always double-check your calculations.

Navigating Unit Conversion

Unit conversion is crucial when measurements are given in different units. In the exercise, we had a measurement of 9 feet 6 inches. To calculate the area correctly in square feet, we converted inches to feet first. This involves knowing that 12 inches make up a foot.
To convert 6 inches to feet, use the conversion factor: \[6 \text{ inches} \div 12 \text{ inches/foot} = 0.5 \text{ feet}\]
This conversion means the total length becomes 9.5 feet. Being comfortable with these conversions is essential because they ensure accuracy in further calculations. Always recheck to confirm no conversion errors, which could alter the entire calculation outcome.

Estimating Paint Coverage

Paint coverage lets us know how much paint is required for a specific area. When we know that 1 pint of paint covers 25 square feet, we can calculate how much paint is needed for a given area by dividing the total area by the coverage rate.
In the problem, we have an area of 90.25 square feet to cover. To find out how many pints are needed, we apply:\[\frac{90.25 \text{ sq ft}}{25 \text{ sq ft/pint}} = 3.61 \text{ pints}\]Since partial pints cannot be bought, and full coverage is necessary, we round up to the nearest whole number, making it 4 pints. Always plan for a little extra if possible, to account for any unforeseen needs or variations in surface type.