Mathematics
Grade 11
15 min
Count coins and bills - up to $5 bill
Count coins and bills - up to $5 bill
What you'll learn
- Identify the value of pennies, nickels, dimes, quarters, and one-dollar bills with 100% accuracy.
- Count a collection of coins (pennies, nickels, dimes, quarters) totaling up to $1.00 and state the total amount with 80% accuracy.
- Solve word problems involving combinations of coins and one-dollar bills, up to a total of $5.00, with at least 70% accuracy.
- Explain how to count a group of mixed coins and one-dollar bills to a classmate, demonstrating understanding of place value and coin values.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Translate real-world constraints about coins and bills into a system of linear inequalities.
Graphically represent a system of monetary inequalities and identify the feasible region of solutions.
Interpret the meaning of integer coordinate pairs within a feasible region in the context of coin and bill combinations.
Formulate an objective function to find the maximum or minimum value of a quantity (e.g., total number of coins) subject to given constraints.
Identify and solve for the vertices of a feasible region to determine optimal solutions.
Analyze and model multi-variable scenarios involving different denominations of currency up to a $5 bill.
You have a mix of quarters and $1 bills in your wallet. You know you have at least 8 items, but the total valu...
2
Key Concepts & Vocabulary
TermDefinitionExample
Linear InequalityA mathematical statement that relates two expressions using an inequality symbol (<, >, ≤, ≥) instead of an equals sign. In our context, it defines a boundary for possible monetary values or quantities.If you have quarters (q) and one-dollar bills (b) with a total value of at most $5, the inequality is `0.25q + 1.00b ≤ 5.00`.
System of Linear InequalitiesA set of two or more linear inequalities containing the same variables. The solution to the system is the set of all ordered pairs that satisfy all the inequalities simultaneously.A wallet has dimes (d) and one-dollar bills (b). There are more than 10 items (`d + b > 10`) and the value is less than $4.00 (`0.10d + 1.00b < 4.00`).
ConstraintA limitation or condition that a solution must sa...
3
Core Formulas
Value Constraint Formula
v_1x_1 + v_2x_2 + ... + v_nx_n \leq T \quad \text{or} \quad v_1x_1 + v_2x_2 + ... + v_nx_n \geq T
Use this to model the total monetary value of a collection. 'v' represents the value of each coin/bill (e.g., 0.25 for a quarter), 'x' is the quantity of that coin/bill, and 'T' is the total value constraint (e.g., at most $5.00).
Quantity Constraint Formula
x_1 + x_2 + ... + x_n \leq N \quad \text{or} \quad x_1 + x_2 + ... + x_n \geq N
Use this to model the total number of items (coins and bills). 'x' represents the quantity of each type of item, and 'N' is the total number of items constraint (e.g., at least 15 items).
Non-Negativity Constraints
x_i \geq 0 \quad \text{for all} \quad i
A fundamen...
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Challenging
A person has a collection of quarters (q) and one-dollar bills (d). The total value is between $3.00 and $5.00, inclusive. There are at least 5 items in total, and the number of quarters is at least double the number of dollar bills. To find the minimum number of coins, what system and objective function should be used?
A.System: 0.25q+d≥3, 0.25q+d≤5, q+d≥5, q≥2d. Objective: Maximize C=q+d.
B.System: 0.25q+d≥3, 0.25q+d≤5, q+d≥5, q≥2d. Objective: Minimize C=q+d.
C.System: 25q+100d=400, q+d≥5, 2q≥d. Objective: Minimize C=0.25q+d.
D.System: 0.25q+d>3, 0.25q+d<5, q+d>5, q>2d. Objective: Minimize C=q+d.
Challenging
A wallet contains nickels (n) and quarters (q) with constraints `n + q ≥ 12` and `0.05n + 0.25q ≤ 2.00`. The feasible region has a vertex at (5, 7). If the total value constraint is tightened to `0.05n + 0.25q ≤ 1.75`, how does this change the feasible region and the vertex?
A.The feasible region shrinks, and the new vertex is (2.5, 9.5).
B.The feasible region expands, and the new vertex is (7.5, 4.5).
C.The feasible region shrinks, and the new vertex is (7.5, 4.5).
D.The feasible region disappears entirely.
Challenging
The optimal solution to a coin problem is found to be 10 dimes and 4 quarters. This combination yields the maximum possible value of $2.00. The solution lies on the boundary lines `d+q=14` and `0.10d+0.25q=2.00`. Which of the following could have been a third constraint in the system?
A.d ≥ 3q
B.q ≥ 5
C.d + q ≥ 15
D.d ≤ 10
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What grade level is "Count coins and bills - up to $5 bill"?
Count coins and bills - up to $5 bill is a Grade 11 Mathematics lesson on ExcelOS.
What will I learn in Count coins and bills - up to $5 bill?
You'll be able to: Identify the value of pennies, nickels, dimes, quarters, and one-dollar bills with 100% accuracy; Count a collection of coins (pennies, nickels, dimes, quarters) totaling up to $1.00 and state the total amount with 80% accuracy….
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How many practice questions are included with Count coins and bills - up to $5 bill?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.