SME
Grade 8 Math · Problem type
When do two plans cost the same?
Use this for “Gym A: $150 to join and $20 a month. Gym B: $60 and $35. When do they cost the same?”
Chart
Plan A: C = 20n + 150
Plan B: C = 35n + 60
They cross at (6, 270).
20 × 6 + 150 = 270
35 × 6 + 60 = 270
This is an example. Type your own values to start a new problem.
C
=
a
n
+
f
C
=
b
n
+
g
n
Months
worked out (not typed)
C
Same amount
worked out (not typed) · $
Formulas
n
=
(
g
−
f
)
÷
(
a
−
b
)
(months
=
(plan
B
to
start
−
plan
A
to
start)
÷
(plan
A
per
month
−
plan
B
per
month))
6 = (60 − 150) ÷ (20 − 35)
C
=
a
×
n
+
f
(same
amount
=
plan
A
per
month
×
months
+
plan
A
to
start)
270 = 20 × 6 + 150
C
=
b
×
n
+
g
(same
amount
=
plan
B
per
month
×
months
+
plan
B
to
start)
270 = 35 × 6 + 60
Assumptions
•
Each plan charges a starting amount plus the same amount every month.
•
Where the lines cross, both cost the same.
•
The plan that starts lower but costs more each month catches up with the other.
Step-by-step
WE KNOW
a = $20 (plan A per month), f = $150 (plan A to start), b = $35 (plan B per month), g = $60 (plan B to start)
FIND
months (n), same amount (C)
Step 1 · Find months (n)
Use n = (g − f) ÷ (a − b) (months = (plan B to start − plan A to start) ÷ (plan A per month − plan B per month))
Set
the
two
right
sides
equal,
gather
n
on
one
side,
then
divide.
n
=
60 − 150
20 − 35
20
n
+
150
=
35
n
+
60
Take
35
n
from
both
sides:
−
15
n
+
150
=
60
Take
150
from
both
sides:
−
15
n
=
−90
Divide
both
sides
by
−15:
n
=
6
n
=
6
Step 2 · Find same amount (C)
Use C = a × n + f (same amount = plan A per month × months + plan A to start)
Put
n
into
the
first
equation.
C
=
20
×
6
+
150
C
=
120
+
150
C
=
$270
Check
Put the numbers back in. Both sides should match.
6 = (60 − 150) ÷ (20 − 35) ✓
270 = 20 × 6 + 150 ✓
270 = 35 × 6 + 60 ✓
Topic check
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