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Notes on arithmetic, rounding, and the strange economics of calculator apps.
How do you find the mode?
The mode is the value that appears most often: in 3, 7, 3, 9, 7, 3 it is 3. Why the answer is the value and not its count, two modes and no mode (and why Python's statistics.mode returns 1 for 1, 3, 5, 8, 13 anyway), mode against median and mean on sick days of 0, 0, 0, 1, 2, 3, 15, the mode of a frequency table, the modal class with its 65 5/11 estimate, and how 0.1 + 0.2 turns one mode into a tie.
How do you find the range?
Subtract the smallest value from the largest: for 4, 9, 2, 15, 7 the range is 15 − 2 = 13. Negative numbers without the sign trap (3 − (−6) is 9, not −3), why one outlier took a set of test scores from a range of 13 to 54 while the median moved one point, the interquartile range, the range of a frequency table (counts do not enter it), and why a computer subtracts 13.1 − 11.9 as 1.1999999999999993.
What is the percent difference formula?
Divide the gap between two values by their average: |a − b| ÷ ((a + b) ÷ 2) × 100, so readings of 72 and 76 are 5.41% apart whichever you read first. Why the average and not one of the values (percent change gives 5.56% one way and 5.26% the other), percent difference against percent change and percent error on the same pair (2.03%, 2.05%, 2.01%), two quotes 11.32% apart, the 200% ceiling, and why a computer subtracts 8.96 − 8.78 as 0.1800000000000015.
How do you calculate mpg?
Divide miles driven by gallons used: fill to full, reset the trip meter, drive, fill to full again, and 312 miles on 11.6 gallons is 26.9 mpg. Averaging several tanks without the trap (30 and 20 mpg can average 26.67, not 25), mpg to L/100km with the exact 235.215 constant, why 10 to 20 mpg saves more fuel than 25 to 50, US versus UK gallons, and cost per mile.
How do you convert kg to lbs?
Divide kilograms by 0.45359237, or multiply by 2.20462: 70 kg is 154.32 lb and 100 kg is 220.46 lb. Pounds and ounces without the 7.7 trap (7.7 lb is 7 lb 11.2 oz, not 7 lb 7 oz), the double it and add ten percent shortcut, stones and pounds, exact lbs to kg tables (150 lb is 68.0388555 kg), and why a computer turns 10 lb into 4.535923700000001 kg.
How do you round decimals?
Find the digit in the place you want and look only at the next digit: 5 or more rounds up, 4 or less leaves it, so 3.14159 to two places is 3.14 and 2.675 is 2.68. Carrying through nines (2.9975 becomes 3.00), the 5 rule versus bankers' rounding (2.5 becomes 2), rounding versus truncating, decimal places versus significant figures, why you round once at the end (3.33 × 3 = 9.99), and why a computer rounds 2.675 to 2.67.
How do you calculate the median?
Sort the values and take the middle one: for 3, 4, 7, 9, 12 the median is 7. With an even count, average the two middle values (2, 5, 8, 11 gives 6.5). The position formula (n + 1) ÷ 2, duplicates and zeros that must stay in, why six house prices have a mean of 536,166.67 and a median of 266,000, the median from a frequency table, and the floating point error hiding in (0.1 + 0.2) ÷ 2.
How do you convert a decimal to a percent?
Multiply by 100, which moves the decimal point two places to the right: 0.375 is 37.5%, 0.04 is 4%, and 0.5 is 50%, not 5%. Decimals above 1 go past 100% (1.5 = 150%), repeating decimals keep their tails (0.333... = 33 1/3%), the way back from percent to decimal (7% = 0.07), and why a calculator turns 0.07 × 100 into 7.000000000000001.
How do you convert a fraction to a percent?
Divide the top by the bottom and multiply by 100: 3/8 is 0.375, which is 37.5%. The scale to 100 shortcut (7/20 = 35/100 = 35%), why 1/3 is 33 1/3% and not 33.33%, fractions past 100%, a table worth memorising, and the way back from percent to fraction.
How do you convert meters to feet?
Multiply by 3.28084, or divide by 0.3048 exactly: 100 m is 328.084 ft. Feet and inches without the 5.9 trap (1.8 m is 5 ft 10.9 in, not 5 ft 9), the times three plus ten percent shortcut, tables in both directions, and the survey foot that was retired in 2022.
What is the simple interest formula?
I = P × r × t: $5,000 at 4% for 3 years earns $600. Worked examples in years, months and days, the units trap, 365 vs 360-day years, solving for the rate or the time, and why a car loan costs far less than P × r × t suggests.
How do you calculate a percentage?
Divide the part by the whole and multiply by 100: 30 out of 120 is 25%. The three question shapes (X% of Y, X is what percent of Y, X is Y% of what), mental shortcuts, the 25% off trap, and why 7 out of 25 prints as 28.000000000000004%.
How do you convert Celsius to Fahrenheit?
Multiply by 1.8 and add 32: 20°C is 68°F, 37°C is 98.6°F. Going back, subtract 32 first, then multiply by 5/9. The double and add 30 shortcut, why a temperature change never gets the 32, and why 72°F is 22.2 repeating.
How do you convert a mixed number to an improper fraction?
Multiply the whole number by the denominator and add the numerator: 3 2/5 = (3 × 5 + 2)/5 = 17/5. Whole numbers, negative mixed numbers, and why you convert before multiplying, so 2 1/2 × 1 3/5 comes out as 4.
How do you convert an improper fraction to a mixed number?
Divide, and the remainder goes over the original denominator: 17/5 = 3 remainder 2 = 3 2/5. Zero remainders, simplifying the fraction part, negative fractions, and why you find the remainder by subtracting, not from the decimal.
How do you simplify fractions?
Divide the top and bottom by their greatest common factor: 18/24 = 3/4. Dividing in small steps, divisibility tests, Euclid's algorithm for 391/667, and why 0.333 as a fraction never simplifies to 1/3.
How do you subtract fractions?
Match the denominators, subtract the tops: 3/4 − 1/6 = 9/12 − 2/12 = 7/12. Borrowing a whole across mixed numbers, the improper fraction escape hatch, and why subtracting rounded decimals is worse than adding them.
How do you divide fractions?
Keep, change, flip: 3/4 ÷ 2/5 becomes 3/4 × 5/2 = 15/8 = 1 7/8. Why the flip is really multiplication, mixed numbers done properly, and why dividing can make the answer bigger.
How do you multiply fractions?
Straight across, no common denominator: 2/3 × 3/4 = 6/12 = 1/2. Cancelling before you multiply, mixed numbers done properly, and why 0.333 × 3 gives 0.999.
How do you convert a repeating decimal to a fraction?
The block goes over that many nines: 0.363636... is 36/99 = 4/11. The algebra behind the rule, repeats that start late, and why 0.999... is exactly 1.
How do you calculate an average percentage?
Not by averaging the percentages: 80% of 20 students and 40% of 5 is 18 of 25, so 72% and not 60%. Parts over wholes, weighting by group size, and when the plain average is safe.
How do you convert a fraction to a decimal?
The fraction bar is a division sign: 3/4 is 3 ÷ 4 = 0.75. Long division by hand, the power of ten shortcut, and why a denominator of 2s and 5s stops while 1/3 never does.
How do you calculate a weighted average?
Value times weight, summed, divided by the total weight: 90, 78 and 85 at 20, 30 and 50 percent give 83.9. GPA by credits, portfolio returns, and the score you need on the final.
How do you calculate an average?
Sum divided by count: 4, 8, 9, 15 gives 36 ÷ 4 = 9. Mean, median and mode, and the five salaries where four people earn under half the average.
How do you convert a decimal to a fraction?
Digits over the matching power of ten, then reduce: 0.75 = 75/100 = 3/4. Repeating decimals go over nines, and 0.33 is not 1/3.
How do you calculate percentage increase?
New minus old, over old, so 1,450 to 1,595 is 10%. The multiplier both ways, increases past 100%, and why three years of raises do not average 8.3333%.
What makes a good fraction calculator app?
Mixed number entry, exact arithmetic instead of decimals underneath, automatic simplifying, and a toggle between 53/12 and 4 5/12. Plus the three second test.
What is the best calculator app for iPhone?
Four questions that decide it: history, whether the math is free, one time price versus weekly, and tracking. With live App Store prices and ratings read on 19 August 2026.
How do you calculate percentage change?
New minus old, divided by old. Increases, decreases, the multiplier shortcut, percentage points, and why down 50% then up 50% leaves you 25% short.
What is scientific notation on a calculator?
A coefficient between 1 and 10, times a power of ten, printed as 6.42E6. Converting both ways, multiplying by adding exponents, the EE key, and engineering notation in threes.
How do you calculate markup?
Profit divided by cost, where margin divides the same profit by the price. The conversion table, keystone pricing, and why marking up 40% leaves you a 28.57% margin.
How do you convert km to miles?
Divide by 1.609344, or multiply by 5 and divide by 8 when there is no calculator to hand. The Fibonacci trick, a table, speed limits, and why the marathon is 42.194988 km.
What does E mean on a calculator?
It is shorthand for a power of ten, so 1.2E8 is 120,000,000. How to read it, how to type one with the EE key, the unrelated lowercase e, and the digits the display quietly drops.
How to calculate a discount percentage
One multiplier does the whole job. The mental route through 10%, how to read the percent off from two prices, why 20% then 10% is 28% and not 30%, and where the cents go missing.
What is the percent error formula?
Subtract, drop the sign, divide by the accepted value, multiply by 100. Worked lab examples, when percent error goes negative, and why it is not percent difference.
How to calculate a tip
Move the decimal one place left, then scale. The mental method for 15, 18 and 20 percent, a rate table, and the cent that binary floating point loses on a 41.30 bill.
How to add fractions
Match the denominators, add the numerators, carry the whole. The cross-multiply shortcut, mixed numbers like 2 3/4 + 1 2/3, and why converting to decimals bakes an error in before you start.
Why does 0.1 + 0.2 = 0.30000000000000004?
Because 0.1 has no exact form in binary, the same way 1/3 has none in decimal. What the machine really stores, why almost every calculator does this, and the arithmetic that does not.
Which calculator apps have no subscription?
Real prices from the live App Store listings. Also the finding we did not expect: the calculator with no ads at all is rated 3.02, and one full of them is rated 4.87.
How to split a bill with tip
An 84.50 bill, 18%, three people. The arithmetic, the mental shortcut, and the direction you have to round so the table does not come up two cents short.