Problem 41
Question
Floor Tiling. The floor of a room is being covered with tile. An area \(\frac{3}{5}\) of the length and \(\frac{3}{4}\) of the width is covered. What fraction of the floor has been tiled?
Step-by-Step Solution
Verified Answer
\[ \frac{9}{20} \]
1Step 1: Identify the Problem
Determine the fraction of the floor that has been tiled by multiplying the fraction of the length that is covered by the fraction of the width that is covered.
2Step 2: Multiply the Fractions
To find the fraction of the floor that is tiled, multiply the fractions of the length and width covered: \ \ \ \ \ \ \[ \frac{3}{5} \times \frac{3}{4} \]
3Step 3: Simplify the Fraction
Multiply the numerators together and the denominators together: \ \ \ \ \ \ \[ \frac{3 \times 3}{5 \times 4} = \frac{9}{20} \]
Key Concepts
Multiplying FractionsSimplifying FractionsArea Calculation
Multiplying Fractions
To find the fraction of the floor that is covered with tiles, it is essential to understand how to multiply fractions. Multiplying fractions involves multiplying the numerators (top numbers) together and the denominators (bottom numbers) together.
For instance, in our exercise, we need to multiply \(\frac{3}{5}\) (fraction of the length covered) by \(\frac{3}{4}\) (fraction of the width covered). The formula for multiplying these fractions is: \[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \] Here, 'a' and 'c' are the numerators, while 'b' and 'd' are the denominators.
For instance, in our exercise, we need to multiply \(\frac{3}{5}\) (fraction of the length covered) by \(\frac{3}{4}\) (fraction of the width covered). The formula for multiplying these fractions is: \[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \] Here, 'a' and 'c' are the numerators, while 'b' and 'd' are the denominators.
Simplifying Fractions
After multiplying the numerators and denominators, the next step is to simplify the fraction, if it is possible. Simplifying fractions ensures they are in their simplest form, making them easier to understand and work with.
Following our example, the product of \(\frac{3}{5} \times \frac{3}{4}\) is \(\frac{9}{20}\). Here’s the detailed breakdown:
\[ 3 \times 3 = 9 \] \[ 5 \times 4 = 20 \] Thus, \[ \frac{3}{5} \times \frac{3}{4} = \frac{9}{20} \] This fraction is already in its simplest form since 9 and 20 have no common factors other than 1.
Following our example, the product of \(\frac{3}{5} \times \frac{3}{4}\) is \(\frac{9}{20}\). Here’s the detailed breakdown:
\[ 3 \times 3 = 9 \] \[ 5 \times 4 = 20 \] Thus, \[ \frac{3}{5} \times \frac{3}{4} = \frac{9}{20} \] This fraction is already in its simplest form since 9 and 20 have no common factors other than 1.
Area Calculation
When understanding how much of the floor has been tiled, we need to calculate the area covered. The area covered by a fraction of the length and width is simply the product of these fractions.
The length fraction covered is \(\frac{3}{5}\) and the width fraction covered is \(\frac{3}{4}\). By multiplying these two fractions, we calculate the fraction of the total area that is tiled. Therefore, \(\frac{3}{5} \times \frac{3}{4} = \frac{9}{20}\), meaning \(\frac{9}{20}\) of the entire floor area is covered.
In summary, the fraction of the floor covered with tiles is \(\frac{9}{20}\). This method is useful for solving numerous real-world problems involving area calculations.
The length fraction covered is \(\frac{3}{5}\) and the width fraction covered is \(\frac{3}{4}\). By multiplying these two fractions, we calculate the fraction of the total area that is tiled. Therefore, \(\frac{3}{5} \times \frac{3}{4} = \frac{9}{20}\), meaning \(\frac{9}{20}\) of the entire floor area is covered.
In summary, the fraction of the floor covered with tiles is \(\frac{9}{20}\). This method is useful for solving numerous real-world problems involving area calculations.
Other exercises in this chapter
Problem 41
Determine whether each number is prime, composite, or neither. $$ 11 $$
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Use \(=\) or \(\neq\) for \(\square\) to write a true sentence. $$ \frac{12}{9} \square \frac{8}{6} $$
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Multiply and simplify. $$ \frac{11}{24} \cdot \frac{3}{5} $$
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Solve. $$ 56+x=194 $$
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