SME
Grade 8 Math · Problem type
Compare two rates
Use this for “One company charges $140 for 5 cubic yards, the other $93 for 3. Which is cheaper per cubic yard?”
Chart
First: y = 28x
Second: y = 31x
Second is steeper: it has the bigger rate, $31 per cubic yard against $28.
This is an example. Type your own values to start a new problem.
m₁
=
y₁
÷
x₁
m₂
=
y₂
÷
x₂
d
Difference per cubic yard, in dollars
worked out (not typed)
Formulas
m₁
=
y₁
÷
x₁
(dollars
per
cubic
yard,
first
=
cost,
first
÷
dirt,
first)
28 = 140 ÷ 5
m₂
=
y₂
÷
x₂
(dollars
per
cubic
yard,
second
=
cost,
second
÷
dirt,
second)
31 = 93 ÷ 3
d
=
m₁
−
m₂
(difference
per
cubic
yard,
in
dollars
=
dollars
per
cubic
yard,
first
−
dollars
per
cubic
yard,
second)
−3 = 28 − 31
Assumptions
•
Each company charges the same amount for every cubic yard: its line goes through (0, 0).
•
Each rate is a slope: cost ÷ cubic yards, from any point on the line.
•
The steeper line costs more for each cubic yard.
Step-by-step
WE KNOW
x₁ = 5 yd³ (dirt, first), y₁ = $140 (cost, first), x₂ = 3 yd³ (dirt, second), y₂ = $93 (cost, second)
FIND
dollars per cubic yard, first (m₁), dollars per cubic yard, second (m₂), difference per cubic yard, in dollars (d)
Step 1 · Find dollars per cubic yard, first (m₁)
Use m₁ = y₁ ÷ x₁ (dollars per cubic yard, first = cost, first ÷ dirt, first)
The first company’s cost for 1 cubic yard.
m₁
=
140
5
m₁
=
$28
Step 2 · Find dollars per cubic yard, second (m₂)
Use m₂ = y₂ ÷ x₂ (dollars per cubic yard, second = cost, second ÷ dirt, second)
The second company’s cost for 1 cubic yard.
m₂
=
93
3
m₂
=
$31
Step 3 · Find difference per cubic yard, in dollars (d)
Use d = m₁ − m₂ (difference per cubic yard, in dollars = dollars per cubic yard, first − dollars per cubic yard, second)
Subtract the rates: the smaller rate is cheaper for each cubic yard.
d
=
28
−
31
$28 < $31, so the first is cheaper
d
=
−3
Check
Put the numbers back in. Both sides should match.
28 = 140 ÷ 5 ✓
31 = 93 ÷ 3 ✓
−3 = 28 − 31 ✓
Topic check
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