5.2 Fractions, Decimals, and Percentages

Why Fractions, Decimals, and Percentages Matter on the PUPCET

These three are the same idea wearing different clothes, and the PUPCET constantly asks you to swap the clothes: Which is largest among 0.6, 5/8, and 62%? What is 3/8 of ₱2,400? Discount, interest, and statistics items in the Mathematics subtest — and even data questions in General Information — all become easy once you convert everything to the form you compute best in.

Operations on Fractions

Adding and subtracting: you need a common denominator. With the same denominator, just add numerators: 2/7 + 3/7 = 5/7. With different denominators, use the least common denominator (LCD):

• 2/3 + 3/4: LCD is 12 → 8/12 + 9/12 = 17/12 = 1 5/12
• 5/6 − 1/4: LCD is 12 → 10/12 − 3/12 = 7/12

Multiplying: multiply straight across, and cancel common factors first to keep numbers small:

• 3/5 × 10/9: cancel 3 into 9 (leaving 3) and 5 into 10 (leaving 2) → (1 × 2)/(1 × 3) = 2/3
• Mixed numbers must become improper fractions first: 1 1/2 × 2 2/3 = 3/2 × 8/3 = 24/6 = 4

Dividing: multiply by the reciprocal (flip the second fraction):

• 4/7 ÷ 2/3 = 4/7 × 3/2 = 12/14 = 6/7

Operations on Decimals

• Add/subtract: align the decimal points and pad with zeros. 3.25 + 0.078 = 3.250 + 0.078 = 3.328
• Multiply: multiply as whole numbers, then count total decimal places. 1.2 × 0.35: 12 × 35 = 420, with 1 + 2 = 3 decimal places → 0.42
• Divide: move the decimal point in the divisor until it is whole, and move the dividend's point the same number of places. 4.5 ÷ 0.15 = 450 ÷ 15 = 30

The Must-Memorize Conversion Table

To convert: fraction → decimal, divide numerator by denominator; decimal → percent, move the point two places right; percent → decimal, move it two places left.

Knowing 1/8 unlocks the whole eighths family: 3/8 is just 3 × 12.5% = 37.5%. That single fact turns many PUPCET comparison items into five-second questions.

Percent of a Number

The word of means multiply. 30% of 250 = 0.30 × 250 = 75. Build speed with anchor percents:

• 10% of any number: move the decimal one place left (10% of ₱480 = ₱48)
• 5%: half of the 10% value (₱24)
• 15%: 10% + 5% (₱72) — perfect for tips and service charges
• 25%: one quarter; 75%: three quarters

Percent Increase and Decrease

Percent change = (new value − original value) ÷ original value × 100%. The base is always the original value — this is where most errors happen.

• Jeepney fare rose from ₱12 to ₱13: increase = 1/12 ≈ 8.33%, not 1/13
• A phone dropped from ₱12,500 to ₱10,000: decrease = 2,500/12,500 = 20%

Discounts in Peso Contexts

Sale price = original price × (1 − discount rate). A ₱1,200 backpack at 25% off costs 1,200 × 0.75 = ₱900. Computing 75% directly is faster than finding 25% and subtracting.

Successive discounts trap: a ₱1,000 jacket at 20% off, then an extra 10% off the sale price: 1,000 × 0.80 = 800, then 800 × 0.90 = ₱720. That is a 28% total discount — NOT 30%, because the second discount applies to the reduced price. Test writers always include the naive sum as a choice.

Simple Interest

I = Prt, where P is the principal, r is the annual rate as a decimal, and t is time in years.

• ₱20,000 deposited at 5% simple interest for 2 years: I = 20,000 × 0.05 × 2 = ₱2,000; total in the account: ₱22,000
• Maria borrows ₱8,000 at 6% per year for 9 months: t = 9/12 = 0.75, so I = 8,000 × 0.06 × 0.75 = ₱360

Watch the time units: months must be converted to years before multiplying.

Common Traps

• Percent of vs. percent more than: 20% of 50 is 10, but 20% more than 50 is 60. Read the exact wording twice.
• The reversal trap: a price that rises 25% then falls 25% does not return to the start. ₱100 → ₱125 → ₱93.75, because the decrease acts on the larger base.
• Decimal shift errors: 0.5% is 0.005, not 0.5. Two-place moves, always.
• Wrong base: after a 20% decrease, getting back needs a 25% increase — the base changed.

Exam Tips

Convert every choice to one form before comparing. For messy percents like 37.5%, check whether the table gives you a fraction (3/8) that makes the multiplication trivial: 37.5% of 800 = 3/8 × 800 = 300. And estimate first — knowing the answer must be near ₱900 lets you eliminate two choices instantly on discount items.