Gear-Ratio GusStoryspinners · word problems

A big gear and a little gear turning together: direct and inverse variation.

Direct and Inverse Variation

Direct and inverse variation. In MathBound, the free Algebra I adventure game, Gear-Ratio Gus asks these problems, and the only way past is to solve them.

Part of More topics: taught by some textbooks beyond the usual course.

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10 questions, no sign-up, nothing saved. Climbs from the basics to harder questions.

What you will practice

Direct Variation. Find the constant of variation (k) and write the equation for a direct variation relationship (y = kx).
Inverse Variation. Find the constant of variation (k) and write the equation for an inverse variation relationship (y = k/x).

Worked examples

Made by the game’s own question engine. Try each one, then open its steps.

Example 1 · Direct Variation

Is

y + 4 = - 2 x

a

direct

variation?

If

it

is,

what

is

the

constant

of

variation

k ?

$Yes:$ $k = - 2$
$Yes:$ $k = - 4$
$No:$ $it$ $cannot$ $be$ $written$ $as$ $y = k x$
$Yes:$ $k = - 6$

Show the worked steps

$Solve$ $for$ $y : y = - 2 x - 4 .$
$The$ $added$ $constant$ $means$ $the$ $line$ $misses$ $the$ $origin,$ $so$ $it$ $is$ $not$ $y = k x .$

Answer: $No:$ $it$ $cannot$ $be$ $written$ $as$ $y = k x$

Example 2 · Inverse Variation

The

table

shows

an

inverse

variation.

What

is

the

missing

value?

x y 148 3 ? 41268

$48$
$3$
$16$
$144$

Show the worked steps

$x y$ $is$ $the$ $same$ $for$ $every$ $column:$ $1 \cdot 48 = 48 .$
$So$ $the$ $missing$ $value$ $is$ $\frac{48}{3} = 16 .$

Answer: $16$

Example 3 · Direct Variation

y

varies

directly

as

x,

and

y = 12

when

x = 3 .

What

is

the

constant

of

variation?

$9$
$36$
$4$
$0.25$

Show the worked steps

$In$ $y = k x, k = \frac{y}{x} .$
$\frac{12}{3} = 4 .$

Answer: $4$

Where it is in your textbook

Textbook	Lessons
Glencoe Algebra 1	3-4, 3-6, 11-1
Holt McDougal Larson Algebra 1	3.6
OpenStax Elementary Algebra 2e	8.9
Pearson Algebra 1 Common Core	5-2, 11-6
SpringBoard Algebra 1	10

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