6.2 Fractions, Decimals, Percentages & Ratios
Key Takeaways
- To find percent change, use the formula: (Change / Original Value) × 100.
- Successive percentage changes cannot simply be added together; you must calculate them sequentially based on the new value.
- Ratios compare parts to parts, while fractions compare parts to the whole.
- When dealing with mixture problems, track the total amount and the amount of the specific substance separately.
Fractions, Decimals, Percentages & Ratios
Fractions, decimals, percentages, and ratios are just four different ways of expressing the same mathematical relationship: a part compared to a whole (or a part compared to another part). The Executive Assessment tests your ability to translate fluently between these formats and use them to solve complex word problems.
Fractions and Decimals
A fraction consists of a numerator (the top number, representing the parts you have) and a denominator (the bottom number, representing the total parts in the whole).
- Addition and Subtraction: You must find a common denominator before adding or subtracting the numerators.
- Multiplication: Multiply numerators together and denominators together.
- Division: Multiply the first fraction by the reciprocal of the second fraction (flip the second fraction).
Decimals are fractions with denominators that are powers of 10. Memorizing common fraction-to-decimal-to-percent conversions will save you significant time on the EA:
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | ≈ 0.333 | ≈ 33.3% |
| 1/4 | 0.25 | 25% |
| 1/5 | 0.2 | 20% |
| 1/6 | ≈ 0.167 | ≈ 16.7% |
| 1/8 | 0.125 | 12.5% |
| 1/10 | 0.1 | 10% |
Percentages and Percent Change
"Percent" literally means "per 100." So, 45% is equivalent to 45/100 or 0.45.
Percent Change is a very common EA topic. The formula is:
Percent Change = (New Value - Original Value) / Original Value × 100 or simply: Percent Change = (Difference / Original Value) × 100
Trap Alert: Always make sure you are dividing by the original value, not the new value. If a stock goes from $40 to $50, the percent increase is 10 / 40 = 25%. If it goes from $50 down to $40, the percent decrease is 10 / 50 = 20%.
Successive Percentage Changes
When a value undergoes multiple percentage changes in a row, you cannot just add the percentages.
Example: A shirt is marked up by 20%, then discounted by 20%. Is it back to its original price? No. Let original price = $100. After 20% markup: $100 × 1.20 = $120. After 20% discount: $120 × 0.80 = $96. The final price is $96, which is a 4% overall decrease.
Simple and Compound Interest
Financial math is a staple of the EA. Be prepared to differentiate between simple and compound interest.
- Simple Interest: Interest is calculated only on the principal amount. Formula: Interest = Principal × Rate × Time (I = PRT)
- Compound Interest: Interest is calculated on the principal AND the accumulated interest from previous periods. Formula: Total Amount = Principal × (1 + Rate)ⁿ, where n is the number of compounding periods.
On the EA, interest problems often require you to approximate or set up the formula rather than compute a messy final decimal. If a problem asks for interest compounded semi-annually, remember to halve the annual rate and double the number of periods.
Ratios and Proportions
A ratio compares the relative sizes of two or more quantities. The ratio of A to B can be written as A:B, A/B, or "A to B".
Crucially, a ratio compares parts to parts, whereas a fraction usually compares parts to a whole. If the ratio of boys to girls in a class is 2:3, then for every 2 boys, there are 3 girls. The total number of "parts" is 2 + 3 = 5. Therefore, boys make up 2/5 of the total class, and girls make up 3/5 of the total class.
A proportion is an equation stating that two ratios are equal: A/B = C/D. You can solve proportions by cross-multiplying: A × D = B × C.
Mixture and Dilution Problems
Mixture problems involve combining two solutions of different concentrations. The most reliable method is to create a table tracking the Total Volume, the Concentration (%), and the Amount of the Solute.
Example: You have 10 liters of a 30% acid solution. How many liters of pure water (0% acid) must you add to dilute it to a 20% acid solution?
| Substance | Volume | Concentration | Amount of Acid |
|---|---|---|---|
| Original Solution | 10 | 30% | 10 × 0.30 = 3 |
| Water Added | w | 0% | w × 0 = 0 |
| New Mixture | 10 + w | 20% | 3 + 0 = 3 |
Set up equation from the New Mixture row: (Amount of Acid) / (Total Volume) = Concentration 3 / (10 + w) = 0.20 3 = 0.20(10 + w) 3 = 2 + 0.20w 1 = 0.20w w = 5 liters.
Mastering these translations and setups will help you breeze through some of the most common quantitative problems on the EA, ensuring you don't get bogged down in arithmetic.
A store increases the price of a television by 25%. A month later, it reduces the new price by 20%. What is the net percentage change from the original price?
If the ratio of cats to dogs at an animal shelter is 3:5, and there is a total of 72 cats and dogs, how many dogs are at the shelter?
How many gallons of a 40% antifreeze solution must be mixed with 10 gallons of a 10% antifreeze solution to produce a 25% antifreeze solution?