Problem 63 - Elementary Algebra Within Reach

Question

Find a ratio that compares the relative sizes of the quantities. (Use the same units of measurement for both quantities.) 3000 pounds to 5 tons

Step-by-Step Solution

Answer

The ratio of 3000 pounds to 5 tons, when converted and simplified, is 3:5.

Step 1 - Convert Tons to Pounds

Since the problem requests a ratio comparison, both quantities should be represented in the same unit. Generally, 1 ton is equal to 2000 pounds. So, first convert 5 tons to pounds. This is done by multiplying the number of tons by the conversion factor, i.e., 5 tons * 2000 pounds/ton.

Step 2 - Create the Ratio

Once the 5 tons have been converted to pounds (in Step 1), the ratio can be formed by expressing the two quantities in a form such as 'a to b' or 'a:b'. So, compare the 3000 pounds to the quantity of pounds obtained from the conversion in Step 1.

Step 3 - Simplify the Ratio

It's often optimal to present the ratio in its simplest form, by reducing it to the lowest possible terms. This is done by finding the greatest common divisor (GCD) of the quantities and dividing both sides of the ratio by this value.

Key Concepts

Units Conversion

When dealing with different units of measurement, one of the key steps is ensuring both quantities are in the same unit. In this case, we have 3000 pounds and 5 tons. Since a ton does not equal a pound, we need to convert tons into pounds to make the comparison. Generally, 1 ton equals 2000 pounds.

• To convert tons to pounds, multiply the number of tons by 2000.
• So, 5 tons becomes 5 x 2000 = 10000 pounds.

This conversion ensures that both quantities are now in pounds, setting a solid foundation to establish the ratio.

Simplifying Ratios

Once both quantities are converted to the same unit, the next step is forming a ratio. Initially, the ratio of 3000 pounds to 10000 pounds appears as 3000:10000. However, to provide a clearer understanding, it's beneficial to simplify this ratio.

• A simplified ratio is easier to interpret and compare.
• It is achieved by dividing both sides of the ratio by their greatest common divisor.

This reduction brings clarity and makes comparisons more meaningful.

Greatest Common Divisor

The greatest common divisor (GCD) is pivotal when simplifying ratios. It is the largest number that divides both quantities without leaving a remainder. For the ratio 3000:10000, we look for the GCD of 3000 and 10000.

• We find that the GCD is 1000, as it evenly divides both numbers.
• By dividing both sides of the ratio by 1000, 3000:10000 simplifies to 3:10.

Using the GCD ensures the ratio is reduced to its simplest form efficiently, providing a neat and easily understandable representation.