Fractions in measurement: choose the operation before calculating
A fraction such as 3/4 describes parts of a reference amount. It does not tell you the operation: are you joining lengths, taking part of a quantity or counting how many pieces fit? The question comes before the calculation.
Addition needs comparable parts
For different denominators, rewrite the fractions using a common subdivision before adding the numerators. Adding denominators directly changes the size of the parts instead of combining their quantities.
First match the measurement unit and the whole being divided. Half a metre and a quarter of a metre can be combined; half a large box and a quarter of a smaller box need the box sizes before they define an amount.
References: OpenStax — Adding and subtracting fractions with different denominators
Multiplication takes a part; division counts parts
Multiplying by 2/3 takes two thirds of an amount. Dividing a length by 1/6 metre asks how many sixth-metre pieces fit. Division uses the reciprocal of a nonzero divisor.
A result greater than 1 is not an error, and a quotient can include part of a piece. If whole pieces are required, separate the complete count from the remainder rather than rounding up.
References: OpenStax — Multiplying, dividing and simplifying fractions
Simplifying does not shrink the amount
Dividing numerator and denominator by a common factor creates an equivalent, easier-to-read fraction. It does not make the quantity smaller. A fraction also preserves values such as 1/3 whose decimal representation never terminates.
For a mixed number such as one and a half, use 3/2 in a calculator expecting numerator/denominator notation. A zero denominator is invalid, and zero has no reciprocal.
References: OpenStax — Multiplying, dividing and simplifying fractions
Worked example: equal pieces of ribbon
Two lengths of the same material are 7/8 metre and 5/12 metre. Assume they can be joined without loss for this arithmetic example. Expressed in twenty-fourths, their total is 21/24 +10/24 =31/24 metre.
For pieces of 1/6 metre, (31/24) ÷ (1/6) =31/4 =7.75. That allows 7 complete pieces with 1/8 metre left over, not 8 complete pieces. Cutting thickness and joining losses are outside this idealised example.