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This essential trains for: SAT-I, GMAT, AMC-8, SAT-II, Math Kangaroo 5-6, Math Kangaroo 7-8.

How do you know if two quantities are vary inversely?

Two quantities vary inversely when their product remains constant as they change.

Example:

A set of food rations lasts a team of 4 climbers for 10 days. If another climber is added to the team, how many days will the rations last?

Assuming there is one ration per day per person, there are:

equation

rations in total.

The number of rations is constant (unchanged).

Denoting the number of days for the larger team by x:

equation

equation

The rations will last 8 days. This stands to reason since there are more people using the same amount of food.

However, a very common error is to mechanically use direct proportionality and to obtain:

equation

equation

which is surely among the answer choices of.

Inversely proportional quantities x and y satisfy the equation of a hyperbola:

equation

figure

To solve problems:

Example:

Quantity x Quantity y
equation equation
equation equation

The product of these quantities remains constant as their values change from the old set of values to the new set of values:

equation

Combined direct/indirect proportionality problems

Example:

Five editors take 3 days to edit 1000 pages. How many days do two editors take to edit 2800 pages?

The number of editors and the number of pages are directly proportional since we assume a constant rate for editing the text.

The number of editors and the number of days needed for a given size of task are inversely proportional.

Place the data in a table (one situation per line, same unit down each column):

equation equation equation
equation equation equation

Identify which columns are inversely proportional and multiply them into a single column:

equation equation
equation equation

Now the table contains directly proportional quantities:

equation

Solve for x to find the number of days:

equation

equation