A rate is a ratio that compares two quantities of different units. It shows how one quantity changes about another. Common examples of rates include speed (distance per time), cost (price per unit), and heart rate (beats per minute).
If a car travels 120 miles in 3 hours, the rate is:
- Rate = 120 miles/3 hours = 40 miles per hour.
This means the car travels 40 miles for every hour of driving.
How to Solve Problems on Rate?
Solving rate problems with fractions involves understanding the relationship between quantities and simplifying or solving fractional expressions. A rate compares two different units, and when fractions are involved, the key steps are simplifying and solving the resulting equation.
Step 1: Identify the quantities being compared and the units involved.
Step 2: Write the rate as a fraction (e.g., speed as distance over time).
Step 3: Multiply or divide fractions as needed to find the unknown quantity.
Step 4: If required, simplify the fraction to its lowest terms for the final solution.
Solved Questions on Rate Problems with Fractions
Question 1: A cyclist covers 3/4 of a mile in 1/2 of an hour. What is the cyclist's speed in miles per hour?
Solution:
We need to find the cyclist's speed, which is the rate of miles per hour. This is given by the formula:
- Speed = Distance/Time = (3/4 miles)/(1/2 hours)
To divide fractions, we multiply the numerator by the reciprocal of the denominator:
- (3/4)/(1/2) = (3/4) × (2/1) = 6/4 = 3/2 = 1.5
So, the cyclist's speed is 1.5 miles per hour.
Question 2: A car travels 7/10 of a mile in 1/5 of an hour. What is the speed of the car in miles per hour?
Solution:
Speed is calculated as: Speed = Distance/Time
Thus, Speed = (7/10)/(1/5) = (7/10) × (5/1) = (7 × 5)/(10 × 1) = 35/10 = 3.5 miles per hour.
The car travels at 3.5 miles per hour.
Question 3: A painter finishes 2/3 of a wall in 1/2 of an hour. What is the painter's rate of work in walls per hour?
Solution:
The rate of work is calculated as:
Rate of Work = (2/3)/(1/2) = (2/3) × (2/1) = (2 × 2)/(3 × 1) = 4/3 walls per hour.
The painter's rate of work is 4/3 walls per hour.
Question 4: A cyclist covers 5/6 of a mile in 1/4 of an hour. What is the cyclist’s speed in miles per hour?
Solution:
The speed is calculated as:
Speed = (5/6)/(1/4) = (5/6) × (4/1) = (5 × 4)/(6 × 1) = 20/6 = 10/3 miles per hour.
The cyclist’s speed is 10/3 miles per hour, or approximately 3.33 miles per hour.
Question 5: A student reads 3/5 of a book in 1/2 of an hour. How many books will the student read in one hour?
Solution:
The rate of reading is:
Rate of Reading = (3/5)/(1/2) = (3/5) × (2/1) = (3 × 2)/(5 × 1) = 6/5 books per hour.
The student reads 6/5 books per hour or 1.2 books per hour.
Worksheet on Rate with Fractions
You can download free worksheet on rate with fractions from below:
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