Question

Express the following ratios in their simplest \(\begin{array}{llll}\text { form: (a) } 6: 3 & \text { (b) } 5: 15 & \text { (c) } 8: 6: 10 & \text { (d) } \frac{1}{2}: \frac{1}{4}\end{array}\) (e) \(\frac{1}{2}: \frac{1}{3}: 1\)

Short answer

Answer: 2:1

Step-by-step solution

Part (a): Simplify the ratio 6:3

To simplify the ratio 6:3, we have to find the GCD of 6 and 3, which is 3. Divide both numbers by the GCD and get the simplified ratio as \(\frac{6}{3}:\frac{3}{3}=2:1\).

Part (b): Simplify the ratio 5:15

To simplify the ratio 5:15, we have to find the GCD of 5 and 15, which is 5. Divide both numbers by the GCD and get the simplified ratio as \(\frac{5}{5}:\frac{15}{5}=1:3\).

Part (c): Simplify the ratio 8:6:10

To simplify the ratio 8:6:10, we have to find the GCD of 8, 6, and 10, which is 2. Divide all three numbers by the GCD and get the simplified ratio as \(\frac{8}{2}:\frac{6}{2}:\frac{10}{2}=4:3:5\).

Part (d): Simplify the ratio \(\frac{1}{2}:\frac{1}{4}\)

First, we need to convert the given fractions to have the same denominator, which is 4 in this case. Rewrite the ratio with the same denominator: \(\frac{2}{4}:\frac{1}{4}\). To simplify, find the GCD of the numerators, which is 1. Divide both numerators by the GCD and we get the simplified ratio as \(\frac{2}{1}:\frac{1}{1}=2:1\).

Part (e): Simplify the ratio \(\frac{1}{2}:\frac{1}{3}:1\)

First, we need to convert the given fractions (and the whole number) to a common denominator, which is 6 in this case. Rewrite the ratio with the same denominator: \(\frac{3}{6}:\frac{2}{6}:\frac{6}{6}\). To simplify, find the GCD of the numerators, which is 1. Divide all numerators by the GCD and we get the simplified ratio as \(\frac{3}{1}:\frac{2}{1}:\frac{6}{1}=3:2:6\).

Key concepts

Simplifying Ratios

Simplifying ratios involves taking a set of numbers and expressing them in the simplest form. When we simplify ratios, we are essentially scaling down the numbers involved, while maintaining their relative sizes to one another.

Here's how you simplify ratios:

• Find the greatest common divisor (GCD) of all numbers in the ratio.
• Divide each number in the ratio by this GCD.

The resulting numbers will represent the same proportion in a simpler form. For example, simplifying the ratio 6:3 means finding the GCD of 6 and 3, which is 3. By dividing both by 3, we get a simplified ratio of 2:1.

In more complex situations, like 8:6:10, you still find the GCD for all the terms. For instance, the GCD is 2, and by dividing each number by 2, you arrive at a ratio of 4:3:5.

Simplification makes ratios more understandable and easier to work with.

Greatest Common Divisor (GCD)

The Greatest Common Divisor, abbreviated as GCD, is the largest integer that can exactly divide two or more numbers without leaving a remainder.

Here's how you find the GCD:

• List all factors of each number.
• Identify the highest number that appears in all lists of factors.

For example, with the numbers 5 and 15, the factors are:

• Factors of 5: 1, 5
• Factors of 15: 1, 3, 5, 15

The largest common factor is 5, so 5 is the GCD.

Using the GCD is crucial for simplifying ratios because it helps reduce the numbers to their smallest possible whole numbers while maintaining their relative proportions.

Fractions to Ratios

Turning fractions into ratios can sometimes seem tricky, but it's easy once you know the steps. First, ensure all fractions have a common denominator.

Follow these steps:

• Convert each fraction so they all have the same denominator.
• Rewrite each number in the ratio based on its numerator.
• Then simplify this new ratio if possible by finding the GCD of the numerators.

For instance, with the fractions \(\frac{1}{2}\) and \(\frac{1}{4}\), we find a common denominator. Here, it's 4:

• \(\frac{1}{2}\) becomes \(\frac{2}{4}\)
• \(\frac{1}{4}\) remains \(\frac{1}{4}\)

This gives us a ratio of 2:1 after observing the numerators only.

The process is similar no matter how many fractions or whole numbers you have, such as with the ratio \(\frac{1}{2}:\frac{1}{3}:1\). Using the common denominator of 6, convert and simplify to result in a ratio of 3:2:6. This makes calculations and comparisons more intuitive.