Mathematics
Grade 11
15 min
Making change
Making change
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Translate a real-world 'making change' scenario into a system of linear inequalities.
Define variables to represent different denominations of coins or bills.
Write inequalities based on constraints like the total number of coins and the total value.
Graph the solution set (feasible region) for a system of inequalities related to making change.
Interpret the vertices and integer points within the feasible region in the context of the problem.
Determine all possible combinations of coins or bills that satisfy a given set of conditions.
Ever been given a handful of change and wondered how many different ways it could add up to the right amount? 🪙 Let's find out using advanced algebra!
This tutorial will show you how to use systems of inequa...
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Key Concepts & Vocabulary
TermDefinitionExample
System of Linear InequalitiesA set of two or more linear inequalities containing the same variables. The solution is the set of all ordered pairs that satisfy all inequalities simultaneously.The set of inequalities `x + y ≤ 10` and `0.05x + 0.10y ≥ 0.50` is a system used to find combinations of nickels (x) and dimes (y).
ConstraintA limitation or condition in a problem that must be satisfied, often expressed as an inequality or equation.The phrase 'you have at most 15 coins' translates to the constraint `x + y ≤ 15`.
Feasible RegionThe solution set of a system of inequalities, represented by the overlapping shaded area on a graph. Every point in this region is a valid solution to the system.When graphing the system for a change-making problem, the feasible...
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Core Formulas
Quantity Constraint
x + y ≤ T or x + y ≥ T or x + y = T
Represents the total number of items (e.g., coins). Let `x` and `y` be the counts of two different denominations and `T` be the total number of items.
Value Constraint
v_1x + v_2y ≤ V or v_1x + v_2y ≥ V or v_1x + v_2y = V
Represents the total monetary value. Let `v_1` and `v_2` be the values of each denomination (e.g., 0.05 for a nickel), `x` and `y` be their counts, and `V` be the total value.
Non-Negativity Constraints
x ≥ 0 and y ≥ 0
Essential for 'making change' problems because the number of coins cannot be negative. This restricts the solution to the first quadrant of the coordinate plane.
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Challenging
Consider two scenarios for making change with dimes (d) and quarters (q). Scenario A: At most 8 coins, value at least $1.00. Scenario B: At most 10 coins, value at least $1.80. Which statement is true about the number of possible integer solutions?
A.Scenario A has more possible integer solutions than Scenario B.
B.Scenario B has more possible integer solutions than Scenario A.
C.Both scenarios have the same number of possible integer solutions.
D.It's impossible to determine without graphing.
Challenging
Given the feasible region defined by d + q ≤ 15, 0.10d + 0.25q ≥ 2.00, d ≥ 0, and q ≥ 0, what is the maximum number of quarters (q) that can be part of a valid solution?
A.8
B.15
C.10
D.12
Challenging
A person claims to have a collection of nickels and dimes that satisfies the following conditions: the total number of coins is no more than 6, and the total value is at least $0.70. Why is this impossible?
A.The boundary lines are parallel and never intersect.
B.The constraints are contradictory, resulting in an empty feasible region.
C.The solution requires a negative number of coins.
D.The maximum possible value with 6 coins is less than $0.70.
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