Converting Between Percentages, Fractions, and Ratios

1. The Interconnected World of Percentages, Fractions, and Decimals

At their core, percentages, fractions, and decimals are simply different ways of representing the same value, specifically a part of a whole. Understanding how to fluidly move between them is the first step to mastering them.

To get a clear picture of what a percentage is and how it relates to fractions and decimals, please watch the following video.

As the video explained, the conversions follow a few simple rules:

• Percent to Fraction: Place the percentage value over 100 and simplify the fraction.
  • Example: $60\%=\frac{60}{100}=\frac{3}{5}$
• Percent to Decimal: Remove the % sign and move the decimal point two places to the left.
  • Example: $60\%=0.60$
• Decimal to Percent: Move the decimal point two places to the right and add the % sign.
  • Example: $0.45=45\%$
• Fraction to Percent: First, convert the fraction to a decimal by dividing the numerator by the denominator. Then, convert the decimal to a percentage.
  • Example: $\frac{1}{4}=0.25=25\%$

2. The Key to Speed: Memorizing Common Conversions

For a time-sensitive exam like the CAT, calculating these conversions every time is too slow. The fastest students commit the most common fraction-to-percentage equivalences to memory. Recognizing these values instantly can turn a complex-looking problem into a simple one.

Let's see a practical example of why this is so important.

The article shows that knowing 71.4% is 5/7 makes the calculation trivial: $\frac{5}{7}\times 119=5\times 17=85$. Without this knowledge, you'd be stuck in a lengthy calculation.

Below is a comprehensive table of these essential conversions. I strongly recommend you spend time memorizing these values. You can save this image or take a screenshot for regular review.

3. Connecting Ratios to the Picture

The final piece of our puzzle is the ratio. A ratio compares two quantities. For example, a ratio of 2:3 means for every 2 units of the first quantity, there are 3 units of the second. This concept is closely linked to fractions.

The following resource explains this relationship and shows how to convert ratios into percentages.

Let's summarize the key conversion process for ratios:

A ratio, such as a:b, usually represents a part-to-part relationship. To express one part as a percentage of the total, you must first find the total.

• Total parts: $a+b$
• Fraction of the first part: $\frac{a}{a+b}$
• Fraction of the second part: $\frac{b}{a+b}$

Once you have the fraction, you can convert it to a percentage as we learned earlier.

Example: The ratio of milk to water in a mixture is 3:2. What is the percentage of milk in the mixture?

1. Ratio: Milk : Water = 3 : 2
2. Total parts: $3+2=5$
3. Fraction of milk: $\frac{\text{Milk}}{\text{Total}}=\frac{3}{5}$
4. Convert to percentage: From our memorized table, we know $\frac{3}{5}=60\%$. So, the mixture contains 60% milk.

4. Mental Math Shortcuts for Any Percentage

While memorizing the table is crucial for common values, you also need a fast way to calculate any percentage. The most effective method is to use 10% and 1% as building blocks.

This video demonstrates the technique very effectively.

The logic is simple and, given your CS background, you can think of it as breaking down a problem into simpler, pre-calculated components.

• To find 10% of any number: Move the decimal point one place to the left.
• To find 1% of any number: Move the decimal point two places to the left.

Example: Calculate 43% of 800.

1. Find 10%: 10% of 800 is 80.
2. Find 1%: 1% of 800 is 8.
3. Build 43%:
   • 40% is $4\times 10\%=4\times 80=320$.
   • 3% is $3\times 1\%=3\times 8=24$.
   • 43% = 40% + 3% = $320+24=344$.

This method is far quicker and less error-prone than the traditional $\frac{43}{100}\times 800$ calculation under exam pressure.

Final Practice!

The image below is a practice worksheet. Try to fill out the first three rows by converting the given value into the other three formats. Use the methods we've learned today.

Show answers for the first three rows

Row 1: Given Ratio 2:5

• Fraction: The total parts are $2+5=7$. This ratio can represent fractions 2/7 or 5/7 depending on the question. If it represents 2 parts out of a total of 5 parts, the fraction is 2/5. Let's assume the latter for this exercise.
• Decimal: $2÷5=0.4$
• Percent: $0.4\times 100=40\%$

Row 2: Given Percent 25%

• Decimal: $0.25$
• Fraction: $25\%=\frac{25}{100}=\frac{1}{4}$
• Ratio: The fraction 1/4 represents 1 part out of a total of 4. This means the other part is $4-1=3$. The part-to-part ratio is 1:3.

Row 3: Given Fraction 4/5

• Decimal: $4÷5=0.8$
• Percent: $0.8\times 100=80\%$
• Ratio: The fraction 4/5 represents 4 parts out of a total of 5. The other part is $5-4=1$. The part-to-part ratio is 4:1.

Conclusion

Excellent work today! We've kicked off the Commercial Mathematics module by mastering a skill that underpins everything to come.

Key Takeaways:

• Percentages, fractions, decimals, and ratios are all interconnected ways of expressing parts of a whole.
• For the CAT, memorizing the common fraction-to-percentage conversion table is essential for speed.
• A ratio a:b can be converted to a fractional part of the whole by using the total a+b as the denominator.
• For any percentage calculation, you can use the 10% and 1% mental math shortcut to build up the value quickly and accurately.

In our next lesson, we will immediately put these skills to use. We will learn to solve problems involving successive percentage changes, profit & loss, and discounts. You will see how a solid grasp of today's conversions makes these word problems much more manageable.