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

How do you know if two quantities are directly proportional?

Two quantities are directly proportional when they depend linearly on one another.

This is to say that, there is a constant rate at which one quantity varies vs. the other quantity.

Also, when one of the quantities is zero, the other must also be zero, i.e. the pairs of values lie on a line that passes through the origin of the system of coordinates.

Typical examples:

Directly proportional quantities Notations How they vary Emphasize the constant rate
Traveling at constant speed, the quantity of fuel used vs. the number of miles traveled. M - Total miles
G - Total gallons
MPG - Miles per gallon
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Amount paid for a set of identical products vs. number of products purchased. TP - Total price
N - Number of items
P - Price per item
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The mass of a homogeneous object vs. its volume. M - Mass
V - Volume
D - Density
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The length of a circular arc vs. the central angle that intercepts it. AL - Arc length
CA - Central angle in radians
R - Radius
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The area of a circular sector vs. the central angle that sweeps it. SA - Sector area
CA - Central angle in radians
R - Radius
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Solve problems that involve direct proportionality by sketching a small table:

Quantity Q is Q1 units large when quantity P is P1 units large
Quantity Q is Q2 units large when quantity P is P2 units large

The amounts in each column must be converted to the same unit (if different units).

The amounts in each line must belong to the same state, moment or situation.

One of the four quantities in the table is not known. Use a proportion to solve for it.

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All possibilities are summarized here (do not memorize, use the properties of proportions!):

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Example:

Two sled dogs eat 2.5 lbs of pemmican per day. How much pemmican does one need to load for a three day trip with 5 dogs?

One dog eats 1.25 lbs of pemmican per day.

One dog eats 3.75 lbs of pemmican in three days.

Five dogs eat:

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Example:

If 5 milliliters of mercury weighs 27.135 grams, how many milliliters does 1 kilogram of mercury occupy?

Convert units (1 kilogram = 1000 grams). Enter the data in the table.

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