Ratios, percentage changes and divisibility

Keep the units on the ratio

A ratio (rapporto) compares two quantities in a stated order. If a mixture contains three parts sand to two parts gravel by mass, the number three describes the sand relative to the gravel's two. A part has no fixed mass until one quantity is given. If the entire mixture weighs thirty kilograms, five equal parts weigh thirty kilograms and each weighs six. The sand accounts for three fifths of the mixture, even though the ratio of sand to gravel is three halves. Write the two labels above the numbers before using either fraction.

Two quantities are directly proportional when multiplying one by a factor multiplies the other by the same factor. Their quotient is then constant. If the product is constant instead, the quantities are inversely proportional. For a fixed length of track, doubling a constant speed halves the travel time. Decide which quantity stays fixed before writing a proportion. Cross multiplication will solve an equation you give it, but cannot tell whether that equation describes the situation.

Record the amount just before the change

A percentage (percentuale) means a number of parts per hundred of some reference quantity. For a rise of one quarter, add the original whole to the extra quarter and multiply by five quarters. For a fall of one quarter, retain three quarters. These multipliers work just as well when the original amount is unknown. Give it a letter and carry the multiplier attached to the letter rather than choosing a convenient amount that might violate another condition.

Successive changes use successive bases. After a rise the reduction is taken from the larger amount. For instance, raising a quantity by a fifth and then lowering it by a fifth gives the multiplier $(6/5)(4/5)=24/25$. The missing twenty-fifth measures the loss relative to the starting amount. To recover a starting value from an ending value, divide by the complete multiplier. Reversing the sign of a percentage rarely reverses the change. Compare a value falling from ten to eight with its return from eight to ten: the same difference of two is a different fraction of each starting value.

Decide whether an integer divides or repeats

A positive integer divides another exactly when the division leaves no remainder. To split two stocks into the greatest possible number of identical batches, that number must divide both stocks. The greatest common divisor (massimo comune divisore) is found by taking the smaller exponent of each shared prime in the two prime decompositions. Every copy of a prime included in the divisor has to be present on both sides. A prime missing on one side contributes nothing.

When two events recur at fixed whole-number intervals after a shared starting time, the first shared positive time has to be a multiple of both intervals. The least common multiple (minimo comune multiplo) takes the larger exponent of each prime from either interval. This time you are assembling a number large enough to contain each interval as a factor. Testing the result by dividing it by both intervals catches missing prime factors. If the events start at different times, a common multiple alone does not account for their different starting positions.

Name what the remainder counts

For a nonnegative number of objects divided into groups of positive size $d$, write $N=dq+r$ with $0\le r<d$. There are $q$ complete groups and $r$ spare objects. If every object must be transported and each vehicle has $d$ places, a nonzero remainder means another vehicle is required. If you are counting full packages available for sale, the same remainder cannot make a further package. Read the noun after the number in your answer. Groups, objects left over and total capacity each correspond to different parts of the same division.