A cashier has a total of 30 bills, made up of ones, fives,...

Name: Problem 102, Chapter 9: Precalculus
Uploaded: 2021-06-02T20:38:52-08:00
Duration: 4 min 43 s
Channel: Lucas Finney
Description: A cashier has a total of 30 bills, made up of ones, fives, and twenties. The number of twenties is 9 more than the number of ones. The total value of the money is 351 dollars. How many of each denomination of bill are there?

A cashier has a total of 30 bills, made up of ones, fives, and twenties. The number of twenties is 9 more than the number of ones. The total value of the money is 351 dollars. How many of each denomination of bill are there?

Key Concepts

Constraint Analysis

Constraint analysis focuses on identifying and applying the specific limitations or conditions provided in a problem to form accurate mathematical equations. It ensures that all the given restrictions, such as the total number of items or relationships between quantities, are properly incorporated into the mathematical model.

Substitution Method

The substitution method is a technique used to solve systems of equations by first solving one equation for one variable, and then substituting that expression into the other equations. This approach reduces the number of variables in the remaining equations, making the system easier to solve.

System of Linear Equations

A system of linear equations is a set of equations in which each equation is linear, meaning it describes a straight-line relationship among variables. Such systems are used to represent problems where multiple linear conditions must be satisfied simultaneously, and solving the system yields the values of the unknown variables.

Algebraic Modeling

Algebraic modeling involves translating the conditions and constraints from a word problem into mathematical expressions and equations. This process is essential for representing real-world scenarios in a form that allows for systematic analysis and solution using algebraic methods.

Transcript

00:01 For this problem, we are told that a cashier has a total of 30 bills made up of ones, fives, and 20s.

00:07 The number of 20s is nine more than the number of ones.

00:10 The total value is $351, and we are asked how much of each denomination are there.

00:16 So this one problem actually is describing a system of equations, even though it's not quite obvious.

00:21 There's the number of ones, the number of fives, the number of 20s.

00:25 Those will be our three unknowns, and we are given three equations.

00:28 So, we know that a plus b plus c, where let's say that a is the number of ones, actually i'll be a little bit more concrete, let's call it o's, f's, and t's, ones, fives, and 20s.

00:47 We know that o plus f plus t is going to equal 30.

00:53 We know that t, the number of 20s, is nine more than the number of ones.

00:59 So t is going to equal o plus 9.

01:04 You have that the total value is $351.

01:08 So that means that o plus 5 times f plus 20 times t equals 351.

01:20 So the second equation, we can plug directly into the first equation and get an expression relating o and t directly, or rather o and f directly.

01:33 So we'd have that o is going to equal, or rather, o plus f plus o plus o plus 9.

01:43 I'll put a little curl in my o to make sure that it doesn't look like a zero.

01:49 So we have o plus f plus o plus 9 is equal to 30...

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