Find an equivalent ratio in simplest terms:
8, colon, 40
8:40
The Correct Answer and Explanation is:
To find an equivalent ratio of 8:40 in its simplest form, follow these steps:
Step 1: Identify the greatest common divisor (GCD)
The first step in simplifying any ratio is to identify the greatest common divisor (GCD) of both numbers. The GCD is the largest number that divides both numbers evenly.
- The factors of 8 are: 1, 2, 4, and 8.
- The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, and 40.
The largest common factor between 8 and 40 is 8. Therefore, the GCD of 8 and 40 is 8.
Step 2: Divide both numbers by the GCD
Now, simplify the ratio by dividing both terms by the GCD (8).
- Divide the first number (8) by the GCD:
( \frac{8}{8} = 1 ) - Divide the second number (40) by the GCD:
( \frac{40}{8} = 5 )
Step 3: Write the simplified ratio
After dividing both numbers by the GCD, the simplified ratio is:
[ 1:5 ]
Explanation of Equivalent Ratios
An equivalent ratio is a ratio that represents the same relationship between numbers but may be expressed in different forms. Ratios can be simplified by dividing both terms by their GCD to make them easier to understand or compare.
For example, the ratio 8:40 means that for every 8 units of the first quantity, there are 40 units of the second quantity. Simplifying this ratio to 1:5 means that for every 1 unit of the first quantity, there are 5 units of the second quantity. Both ratios represent the same relationship, but the simplified ratio 1:5 is easier to interpret.
In everyday situations, simplifying ratios makes it easier to work with proportions, scale models, recipes, and other applications where relationships between quantities need to be compared or scaled.
Thus, the equivalent ratio in simplest form of 8:40 is 1:5.