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Work through fractions, percentages, measurements, shapes, and useful calculations with examples.
Search this topicMultiply the whole number by the denominator, add the numerator, and keep the denominator. For example, 4 2/5 becomes (4 × 5 + 2)/5 = 22/5. The value has not changed: four wholes contain twenty fifths, with two more fifths left over. This form is useful before multiplying mixed measurements.
Divide the numerator by the denominator. The whole-number quotient is the whole part, and the remainder becomes the numerator over the original denominator. For 29/6, 29 ÷ 6 gives 4 remainder 5, so the result is 4 5/6. Check by converting back: 4 × 6 + 5 = 29.
Multiply or divide the numerator and denominator by the same nonzero number. For instance, 3/7 and 12/28 are equal because both parts were multiplied by four. Changing only one part changes the value. Equivalent fractions let you express the same amount using smaller or larger equal-sized pieces.
Divide its numerator and denominator by a common factor, repeating until no factor greater than one remains. For 24/36, dividing both by twelve gives 2/3. You can reach the same result in smaller stages. Reduce factors, not individual digits; crossing matching digits out of unrelated numbers does not preserve the fraction.
Multiply the numerators together and the denominators together, then simplify. For 3/5 × 5/8, the result is 15/40 = 3/8. Common factors can be canceled before multiplication to keep the arithmetic small. Unlike addition, multiplication does not require matching denominators; do not add the denominators or cross-multiply diagonally.
Keep the first fraction and multiply by the reciprocal of the second. For 5/6 ÷ 2/3, calculate 5/6 × 3/2 = 15/12 = 1 1/4. Only the divisor is inverted. The divisor cannot be zero, and any mixed numbers should first be rewritten as improper fractions.
Convert both mixed numbers to improper fractions before multiplying. For 2 1/2 × 1 3/5, calculate 5/2 × 8/5 = 4. Multiplying only the whole parts and then the fractions misses cross terms. Keep units attached: multiplying two lengths produces an area, while multiplying a length by a plain quantity produces another length.
Treat the main fraction bar as division and group everything above it separately from everything below it. For (3/4)/(5/8), calculate 3/4 ÷ 5/8 = 6/5. Simplify any sums in each group first. On a calculator, parentheses help preserve those groups rather than accidentally changing the order.
Add or subtract the numerators and keep the denominator, then simplify if possible. For 5/9 + 2/9, the result is 7/9; for 5/9 − 2/9, it is 3/9 = 1/3. The denominator describes the size of each piece, so it stays unchanged when counting more or fewer identical pieces.
Find a number divisible by both denominators; the smallest such number keeps the arithmetic simplest. For denominators six and eight, use twenty-four. Multiply both parts of each fraction as needed: 1/6 becomes 4/24 and 3/8 becomes 9/24. Multiplying the denominators also works, but may create unnecessarily large numbers.
Rewrite both as equivalent fractions with a common denominator, then add their numerators. For 1/3 + 1/4, use twelfths: 4/12 + 3/12 = 7/12. Do not add the original denominators to get 2/7. Check the size: the sum should be larger than either positive fraction you started with.
Convert to a common denominator first, then subtract the numerators. For 7/8 − 1/3, use twenty-fourths: 21/24 − 8/24 = 13/24. Keep the subtraction order unchanged, because reversing it changes the sign. Simplify the final fraction only if numerator and denominator still share a common factor.
Use a common denominator, add the whole parts and fraction parts, then carry any whole amount from the fraction. For 2 3/4 + 1 5/8, rewrite three quarters as six eighths. The sum is 3 11/8 = 4 3/8. Converting everything to improper fractions first is also valid.
Rewrite one whole from the first number as a fraction with the common denominator. For 5 1/4 − 2 3/4, change 5 1/4 to 4 5/4. Then subtract to get 2 2/4 = 2 1/2. The regrouping changes the form of the first measurement, not its value.
Write the decimal digits over the matching place-value denominator and simplify. For 0.48, write 48/100, then reduce to 12/25. Two decimal places mean hundredths; three mean thousandths. Preserve any whole-number part and negative sign. This method applies to a decimal that ends, not an endless repeating pattern.
Line up the decimal points and add trailing zeros where helpful, then compare place values from left to right. For 0.7 and 0.65, compare 0.70 with 0.65: seventy hundredths is larger. More written decimal digits do not automatically mean a larger value. Negative values need their position on the number line considered.
Locate the last place you want to keep and inspect the next digit. Under ordinary school rounding, five or more raises the kept digit; below five leaves it unchanged. Thus 6.278 rounds to 6.28 at two decimal places. Round at the end of a calculation when possible to avoid accumulating small errors.
Write the numbers vertically with their decimal points aligned, filling missing places with zeros if helpful. Then work column by column as with whole numbers. For 7.4 + 0.86, use 7.40 + 0.86 = 8.26. Align place values rather than the rightmost digits, and keep the answer’s decimal point in that column.
Multiply as whole numbers, then give the result the total number of decimal places in both factors. For 1.4 × 0.3, calculate 14 × 3 = 42 and place two decimal digits: 0.42. Estimate first to catch errors; three tenths of 1.4 should be smaller than 1.4.
Multiply both numbers by the same power of ten until the divisor is a whole number. For 4.2 ÷ 0.6, change it to 42 ÷ 6 = 7. Moving only one decimal point changes the problem. Continue ordinary division and check by multiplying the answer by the original divisor.
Divide the numerator by the denominator. For 7/16, calculate 7 ÷ 16 = 0.4375. The denominator goes outside the long-division bracket; reversing the numbers gives a different value. Some fractions produce repeating decimals, so keep the fraction or indicate approximation if you round a result that does not terminate.
A repeated remainder creates a repeated block of decimal digits. For example, 2/9 = 0.222… and never ends. A finite display such as 0.2222 is an approximation, not the entire exact decimal. Keep the fraction for exact work, or write the repeating notation and state how many decimal places you need.
Divide by one hundred and remove the percent sign. For example, 7.5% becomes 0.075, while 140% becomes 1.4. The latter is larger than one because it describes more than the whole reference amount. To reverse the conversion, multiply the decimal by one hundred and add the percent sign.
Divide the numerator by the denominator, then multiply by one hundred. For 9/20, calculate 9 ÷ 20 = 0.45, which is 45%. If the division repeats, retain enough precision before rounding the percentage. A fraction greater than one correctly gives a percentage greater than one hundred.
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