Category: Math

  • How to Calculate a Discount: Formulas, Examples & Stacked Discounts

    Discounts are everywhere — sales, clearance events, coupon codes, wholesale pricing, and subscription renewals. Knowing how to calculate a discount quickly means you will never overpay, and you will always know whether a deal is actually a deal. This guide covers every discount scenario you will encounter, with formulas, worked examples, and tips for spotting misleading discount claims.

    How to Calculate a Discount: The Basic Formula

    Discount Amount = Original Price × (Discount % ÷ 100)
    Sale Price = Original Price − Discount Amount

    Example: A jacket costs $120 and is 35% off. Discount = $120 × 0.35 = $42. Sale price = $120 − $42 = $78.

    How to Find the Discount Percentage

    If you want to calculate what percentage off a sale price represents: Discount % = ((Original − Sale) ÷ Original) × 100.

    Example: A blender was $80, now $60. Discount % = ((80 − 60) ÷ 80) × 100 = (20 ÷ 80) × 100 = 25% off.

    How to Find the Original Price

    If you know the sale price and the discount percentage, work backward: Original Price = Sale Price ÷ (1 − Discount/100).

    Example: Shoes are $75 after a 25% discount. Original = 75 ÷ (1 − 0.25) = 75 ÷ 0.75 = $100.

    Discount Quick Reference Table

    Original Price10% Off20% Off30% Off40% Off50% Off
    $50$45$40$35$30$25
    $100$90$80$70$60$50
    $150$135$120$105$90$75
    $200$180$160$140$120$100
    $500$450$400$350$300$250

    How to Calculate Stacked Discounts

    When a retailer offers multiple discounts (e.g., 20% off, then an additional 10% coupon), they apply sequentially, not additively. A 20% discount followed by a 10% discount is not the same as a 30% discount.

    Example: A $200 item is 20% off (= $160), then you apply a 10% coupon. The 10% applies to $160, not $200: 10% of $160 = $16. Final price = $160 − $16 = $144. This is a 28% total discount, not 30%. To find the combined discount: 1 − (0.80 × 0.90) = 1 − 0.72 = 28%.

    How to Spot Fake Discounts

    Some retailers inflate the “original price” before marking it down — a practice known as “reference price inflation.” The item may have never actually sold at the stated original price, making the discount misleading. Before trusting a sale price, check the item’s price history using tools like CamelCamelCamel (for Amazon), Google Shopping, or browser extensions that track historical prices. A genuine 40% discount looks very different from a price that was quietly raised last week just to be “marked down” today.

    Use Our Free Discount Calculator

    Find the exact sale price and your savings instantly with our free Discount Calculator. Enter the original price and discount percentage to get the result in one click.

    Frequently Asked Questions

    Does sales tax apply before or after the discount?

    In most jurisdictions, sales tax is calculated on the discounted price (after the discount is applied). This means you save on both the item price and the tax. For example, a $100 item with a 20% discount and 8% tax: sale price = $80, tax = $80 × 0.08 = $6.40, total = $86.40.

    What is the difference between a discount and a rebate?

    A discount reduces the price at the point of sale — you pay less immediately. A rebate is a partial refund submitted after the purchase. Rebates require effort (mailing forms or submitting receipts online) and have an expiration date. Studies show that 20–40% of rebates go unclaimed, which is why retailers use them — the advertised savings are not always realized by the buyer.

    Is a “Buy One Get One Free” the same as 50% off?

    No. “Buy one get one free” means you get 2 items for the price of 1 — an effective discount of 50% per item only if you were going to buy two anyway. If you only wanted one item, it is not actually a 50% saving because you are spending more than you originally planned. A true 50% discount means each individual item is half price regardless of quantity.

  • How to Calculate Percentage Increase or Decrease (Formula + Examples)

    Whether you are analyzing sales growth, tracking price changes, monitoring investment returns, or comparing test scores, knowing how to calculate percentage increase or decrease is a fundamental everyday skill. This guide covers every scenario with clear formulas, real-world examples, and a reference table — plus the common mistakes that trip people up.

    Percentage Increase Formula

    % Increase = ((New Value − Old Value) ÷ Old Value) × 100

    Example: A product was $40 and is now $52. % Increase = ((52 − 40) ÷ 40) × 100 = (12 ÷ 40) × 100 = 30% increase.

    Percentage Decrease Formula

    % Decrease = ((Old Value − New Value) ÷ Old Value) × 100

    Example: Revenue dropped from $80,000 to $64,000. % Decrease = ((80,000 − 64,000) ÷ 80,000) × 100 = (16,000 ÷ 80,000) × 100 = 20% decrease.

    More Worked Examples

    Example 1 — Salary increase: You earned $55,000 last year and received a raise to $60,500. % increase = ((60,500 − 55,000) ÷ 55,000) × 100 = (5,500 ÷ 55,000) × 100 = 10%.

    Example 2 — Stock price drop: A share fell from $148 to $111. % decrease = ((148 − 111) ÷ 148) × 100 = (37 ÷ 148) × 100 = 25%.

    Example 3 — Population growth: A city grew from 250,000 to 312,500 residents over a decade. % increase = ((312,500 − 250,000) ÷ 250,000) × 100 = 25%.

    Example 4 — Finding the new value from a percentage change: A car worth $24,000 depreciates by 15% in its first year. New value = 24,000 × (1 − 0.15) = 24,000 × 0.85 = $20,400.

    Percentage Change Reference Table

    ScenarioOld ValueNew Value% Change
    Price rise$100$125+25%
    Price drop$100$75−25%
    Salary increase$50,000$55,000+10%
    Weight loss200 lbs180 lbs−10%
    Revenue growth$1M$1.3M+30%

    Common Mistakes to Avoid

    Using the wrong base value: Always divide by the original (old) value, not the new one. A price that rises from $80 to $100 is a 25% increase — not 20%. The 20% figure comes from dividing by the new value ($100), which is incorrect.

    Confusing percentage points with percentage change: If an interest rate rises from 3% to 5%, it increases by 2 percentage points — but by 66.7% in relative terms. These are different measurements and are frequently confused in news reporting and financial documents.

    Assuming increases and decreases cancel out: A 25% increase followed by a 25% decrease does not return to the original value. If something rises from $100 to $125 (+25%), then falls 25% from $125, it lands at $93.75 — not $100. The decrease applies to a higher base.

    How to Find the Original Value Before a Percentage Change

    If you know the final value and the percentage change, you can work backward to find the original: Original = New Value ÷ (1 + % change/100) for an increase, or Original = New Value ÷ (1 − % change/100) for a decrease.

    Example: A price is now $90 after a 10% decrease. Original = 90 ÷ (1 − 0.10) = 90 ÷ 0.90 = $100.

    Use Our Free Percentage Calculator

    For quick calculations, use our free Percentage Calculator. It handles the most common percentage tasks — finding X% of a number, calculating percentage change, and more — without any sign-up required.

    Frequently Asked Questions

    How do I calculate percentage increase between two numbers?

    Subtract the old value from the new value, divide by the old value, then multiply by 100. The result is the percentage increase. If the result is negative, it is a decrease rather than an increase.

    What if the old value is zero?

    If the original value is zero, the percentage change formula is undefined — you cannot divide by zero. In this case, it is more meaningful to simply state the absolute change (e.g., “increased from 0 to 50 units”) rather than express it as a percentage.

    Can percentage change exceed 100%?

    Yes. A 100% increase means the value doubled. A 200% increase means it tripled. There is no mathematical upper limit to percentage increase. For decreases, however, the maximum is 100% — a value cannot decrease by more than 100% of itself (that would bring it below zero, which may or may not be meaningful depending on context).

  • How to Calculate Percentage: 5 Types Every Person Should Know

    Percentages show up everywhere in daily life — sales discounts, tax rates, exam scores, salary raises, nutrition labels, and investment returns. Yet many people freeze when faced with a percentage calculation that is not straightforward. This guide covers the five most common percentage calculations, with clear formulas and real-world examples you can use immediately.

    What Is a Percentage?

    A percentage is a way of expressing a number as a fraction of 100. The word “percent” comes from the Latin per centum, meaning “by the hundred.” So 45% simply means 45 out of every 100 — or 0.45 as a decimal, or 45/100 as a fraction.

    The key to working with percentages is converting between three forms: the percentage itself (45%), the decimal (0.45), and the fraction (9/20). Most calculations become easy once you are comfortable moving between these forms.

    Type 1: What Is X% of Y?

    This is the most common percentage calculation. The formula is simple:

    Result = (X / 100) × Y

    Example: What is 15% of $240?
    Result = (15 / 100) × 240 = 0.15 × 240 = $36

    Real-world uses: calculating a tip, finding a sale discount amount, or determining how much tax you will pay on a purchase.

    Mental math shortcut: To find 10% of any number, just move the decimal point one place to the left. Then multiply or divide as needed. 10% of $240 = $24. So 15% = $24 + $12 (half of $24) = $36.

    Type 2: X Is What Percent of Y?

    This finds the percentage that one number represents of another. The formula is:

    Percentage = (X / Y) × 100

    Example: You scored 78 out of 120 on a test. What percentage did you get?
    Percentage = (78 / 120) × 100 = 0.65 × 100 = 65%

    Real-world uses: calculating your test score, finding what percentage of your budget you spent on food, or determining a completion rate on a project.

    Type 3: Percentage Increase

    Percentage increase measures how much a value has grown relative to its starting point. The formula is:

    Percentage Increase = ((New Value − Old Value) / Old Value) × 100

    Example: Your rent went from $1,200 to $1,380. What is the percentage increase?
    Percentage Increase = ((1,380 − 1,200) / 1,200) × 100 = (180 / 1,200) × 100 = 15%

    Real-world uses: evaluating salary raises, tracking investment growth, calculating inflation, or comparing prices year over year.

    Type 4: Percentage Decrease

    The mirror image of percentage increase. The formula is:

    Percentage Decrease = ((Old Value − New Value) / Old Value) × 100

    Example: A jacket that cost $150 is now on sale for $105. What is the discount percentage?
    Percentage Decrease = ((150 − 105) / 150) × 100 = (45 / 150) × 100 = 30%

    Real-world uses: evaluating sales and discounts, tracking price drops, measuring weight loss, or calculating how much crime rates or other statistics changed.

    Type 5: Finding the Original Value Before a Percentage Change

    Sometimes you know the final value and the percentage change, and you need to work backward to find the original. This trips people up frequently.

    Formula for original value before a percentage increase:
    Original = New Value / (1 + percentage/100)

    Formula for original value before a percentage decrease:
    Original = New Value / (1 − percentage/100)

    Example: A shirt costs $85 after a 15% discount. What was the original price?
    Original = 85 / (1 − 0.15) = 85 / 0.85 = $100

    Common mistake: Many people add 15% back to $85 and get $97.75. That is wrong. You need to divide by the remaining percentage, not add the discount back.

    Percentage vs. Percentage Points: An Important Distinction

    This is one of the most misunderstood concepts in everyday math and finance:

    • If the interest rate goes from 4% to 5%, that is an increase of 1 percentage point.
    • But in percentage terms, it is an increase of 25% (because 1 is 25% of 4).

    Politicians, media, and marketers frequently blur this distinction — sometimes intentionally. A politician might say taxes increased by “only 2 percentage points” when the actual percentage increase is much larger. Always ask: is this a percentage point change or a percentage change?

    Quick Reference: Percentage Formulas

    • X% of Y: (X ÷ 100) × Y
    • X is what % of Y: (X ÷ Y) × 100
    • % increase: ((New − Old) ÷ Old) × 100
    • % decrease: ((Old − New) ÷ Old) × 100
    • Original before % increase: New ÷ (1 + %/100)
    • Original before % decrease: New ÷ (1 − %/100)

    Common Percentage Mistakes to Avoid

    Mistake 1 — Confusing percent change with percentage point change. As explained above, these are different. A rate going from 2% to 3% is a 1 percentage-point increase but a 50% relative increase.

    Mistake 2 — Adding percentages directly. If a price increases by 20% and then decreases by 20%, you do NOT end up where you started. A $100 item increases to $120, then decreases by 20% to $96. You lost $4.

    Mistake 3 — Using the wrong base. Always clarify: percentage of what? A 30% profit margin calculated on revenue is very different from 30% calculated on cost.

    Use the Percentage Calculator

    For quick answers without the mental math, our free Percentage Calculator handles all five types of calculations instantly. Just enter your numbers and get results in seconds — no account required.

    Frequently Asked Questions

    How do I calculate 20% off a price?
    Multiply the original price by 0.20 to get the discount amount, then subtract it. Or simply multiply by 0.80 to get the sale price directly. Example: 20% off $75 = $75 × 0.80 = $60.

    How do I calculate a percentage on a calculator?
    Most calculators have a % button. To find 15% of 200: press 200 × 15 %. Alternatively, just calculate 200 × 0.15 = 30.

    What percentage is 3 out of 20?
    (3 ÷ 20) × 100 = 15%.

    How do I increase a number by a percentage?
    Multiply the number by (1 + percentage/100). To increase 500 by 12%: 500 × 1.12 = 560.