SME
Grade 9 Math · Problem type
A budget and a count
Use this for “Adult tickets are $10 and student tickets $6. Spend at most $144 on at most 20 tickets.”
Chart
Money: s ≤ (−5/3)a + 24
Count: s ≤ −a + 20
Where both sides are shaded, both inequalities are true: those points are the solutions.
Both lines are solid: points on them are included.
The boundaries cross at (6, 14)
Test (5, 12)
−5/3 × 5 + 24 = 47/3, and 12 ≤ 47/3 is true
−1 × 5 + 20 = 15, and 12 ≤ 15 is true
It is a solution.
This is an example. Type your own values to start a new problem.
$
A
a
+
$
S
s
≤
$
M
a
+
s
≤
N
a₀
Adult tickets to test
example
s₀
Student tickets to test
example
m₁
Slope of the money line
worked out (not typed)
i₁
Money line’s s-intercept
worked out (not typed)
a
Corner x: where the lines cross
worked out (not typed)
s
Corner y: where the lines cross
worked out (not typed)
C
Test cost
worked out (not typed) · $
n
Test count
worked out (not typed)
Formulas
m₁
=
−
A
S
−
5
3
=
−
10
6
i₁
=
M
S
24
=
144
6
a
=
M
−
S
×
N
A
−
S
6
=
144 − 6 × 20
10 − 6
s
=
N
−
a
14 = 20 − 6
C
=
A
×
a₀
+
S
×
s₀
122 = 10 × 5 + 6 × 12
n
=
a₀
+
s₀
17 = 5 + 12
Assumptions
•
One inequality for the money, Aa + Ss ≤ M, and one for the count, a + s ≤ N.
•
Solve each for s to graph it; both shade below their lines.
•
Only whole numbers of tickets, 0 or more, make sense.
Step-by-step
WE KNOW
A = $10, S = $6, M = $144, N = 20, a₀ = 5, s₀ = 12
FIND
slope of the money line (m₁), money line’s s-intercept (i₁), corner x: where the lines cross (a), corner y: where the lines cross (s), test cost (C), test count (n)
Step 1 · Find slope of the money line (m₁)
Use m₁ = −A ÷ S
Solve
Aa
+
Ss
≤
M
for
s
:
take
Aa
from
both
sides
and
divide
by
S
.
This
is
the
slope.
m₁
=
−
10
6
m₁
=
−
5
3
Step 2 · Find money line’s s-intercept (i₁)
Use i₁ = M ÷ S
And
the
s
-intercept:
all
student
tickets.
i₁
=
144
6
i₁
=
24
Step 3 · Find corner x: where the lines cross (a)
Use a = (M − S × N) ÷ (A − S)
Where
the
lines
cross:
put
s
=
N
−
a
into
Aa
+
Ss
=
M
,
so
(
A
−
S
)
a
=
M
−
SN.
a
=
144 − 6 × 20
10 − 6
a
=
144 − 120
4
a
=
24
4
a
=
6
Step 4 · Find corner y: where the lines cross (s)
Use s = N − a
On the count line, the rest of the tickets are student tickets.
s
=
20
−
6
s
=
14
Step 5 · Find test cost (C)
Use C = A × a₀ + S × s₀
Test
the
money
inequality:
the
cost
of
the
test
tickets
must
be
at
most
M
.
C
=
10
×
5
+
6
×
12
C
=
50
+
72
C
=
$122
Step 6 · Find test count (n)
Use n = a₀ + s₀
Test
the
count:
at
most
N
tickets.
n
=
5
+
12
n
=
17
→
both
are
true:
the
test
point
is
a
solution
Check
Put the numbers back in. Both sides should match.
−
5
3
=
−
10
6
✓
24
=
144
6
✓
6
=
144 − 6 × 20
10 − 6
✓
14 = 20 − 6 ✓
122 = 10 × 5 + 6 × 12 ✓
17 = 5 + 12 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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