Question

Kim drew \(\frac{5}{8}\) of a picture in \(\frac{5}{12}\) of an hour. What is her picture-drawing rate in pictures/ hour? A. \(\frac{2}{3}\) picture/hour B. \(\frac{3}{4}\) picture/hour C. \(1 \frac{1}{2}\) pictures/hour D. \(1 \frac{3}{4}\) pictures/hour

Short answer

Option C: \(1 \frac{1}{2}\) pictures/hour.

Step-by-step solution

Understand the Problem

Kim drew \(\frac{5}{8}\) of a picture in \(\frac{5}{12}\) of an hour. The goal is to find her picture-drawing rate in pictures per hour.

Set Up the Rate Equation

To find the rate, use the formula: \(\text{Rate} = \frac{\text{Amount of work}}{\text{Time}}\). Here, the amount of work is \(\frac{5}{8}\) of a picture and the time is \(\frac{5}{12}\) of an hour.

Write Down the Fraction Division

The rate is \(\frac{\frac{5}{8}}{\frac{5}{12}}\). Solving this involves dividing two fractions.

Divide the Fractions

To divide fractions, multiply the first fraction by the reciprocal of the second fraction: \(\frac{5}{8} \times \frac{12}{5}\).

Simplify the Multiplication

Multiply the numerators together and the denominators together: \(\frac{5 \times 12}{8 \times 5} = \frac{60}{40}\).

Simplify the Resulting Fraction

Simplify \(\frac{60}{40}\) to its lowest terms. Divide both the numerator and the denominator by their greatest common divisor (20): \(\frac{60}{40} = \frac{3}{2}\).

Convert to Mixed Number

Convert \(\frac{3}{2}\) to a mixed number. \(\frac{3}{2} = 1 \frac{1}{2}\).

Choose the Correct Option

The correct rate in pictures per hour is \(\boxed{1 \frac{1}{2}}\) pictures/hour, which corresponds to Option C.

Key concepts

Rate Calculation

In this exercise, we need to calculate Kim's picture-drawing rate. A rate is a ratio comparing two quantities of different units. In this case, Kim's rate is the amount of picture she drew per hour.
To determine the rate, we can use the formula: \(\text{Rate} = \frac{\text{Amount of work}}{\text{Time}}\). Here, the amount of work is the fraction of the picture Kim drew, and the time is the fraction of an hour she took to draw it.
By setting up the rate equation correctly, we can accurately find the rate in pictures per hour.

Fractions

Fractions represent parts of a whole. They consist of a numerator (top number) and a denominator (bottom number).
In the problem, Kim completed \(\frac{5}{8}\) of a picture, meaning she did 5 parts out of 8 parts of the picture.
Similarly, \(\frac{5}{12}\) of an hour means she took 5 parts out of 12 parts of an hour. Understanding these fractions is crucial for setting up and solving the rate equation.

Mathematical Reasoning

Solving this problem involves logical steps to break down the mathematical reasoning. Starting with understanding the problem, we identify the quantities involved and determine what's being asked. We convert the word problem into a mathematical equation.
Next, we handle fraction division by recognizing that dividing by a fraction is the same as multiplying by its reciprocal. This key reasoning lets us transform \(\frac{\frac{5}{8}}{\frac{5}{12}}\) into \(\frac{5}{8} \times \frac{12}{5}\).
Finally, simplifying the multiplication and converting to a mixed number completes the reasoning process to arrive at the solution.

Simplifying Fractions

After multiplying fractions, simplifying the resulting fraction helps make the answer more understandable. Simplicity involves reducing a fraction to its lowest terms.
In this exercise, after multiplying, we get \(\frac{60}{40}\). To simplify, we find the greatest common divisor (GCD) of 60 and 40, which is 20. We divide both the numerator and the denominator by this GCD:
\(\frac{60}{40} = \frac{60 \.div 20}{40 \.div 20} = \frac{3}{2}\).
Finally, converting \(\frac{3}{2}\) to the mixed number, 1 \(\frac{1}{2}\), makes the result clear and easy to interpret.