Value of Coins Mary has 3.00 in nickels, dimes, and quarters. If she has twice as many dimes as

step 1 All right, so we have Mary has $3 in nickels, dimes, quarters. Let's follow this part by part. S

Value of Coins Mary has 3.00 in nickels, dimes, and...

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple equations with multiple variables. Solving these systems allows one to find the values of unknowns that satisfy all of the given conditions simultaneously, such as the total monetary value and the relationships between different coins.

Translation of Word Problems

Translating a word problem entails converting the descriptive conditions and relationships into mathematical equations. For coin problems, this means turning worded constraints, like having twice as many dimes as quarters, into algebraic expressions that can be solved systematically.

Substitution Method

The substitution method is a technique used to solve systems of equations by expressing one variable in terms of another and then substituting this expression into the remaining equations. This method simplifies the problem by reducing the number of unknowns step by step until a solution is found.

Coin Combination Problems

These problems involve determining the number of coins of different denominations that add up to a specific total value. The approach typically involves understanding the value of each coin type and how their quantities influence the overall sum.

Variable Assignment

This concept involves representing unknown quantities with variables, which is essential in setting up equations. In the context of coin problems, each type of coin is assigned a variable to denote its count, facilitating the process of mathematical modeling.

Transcript

00:01 All right, so we have mary has $3 in nickels, dimes, quarters.

00:04 Let's follow this part by part.

00:06 She has twice as many dimes as quarters.

00:09 So let's say quarters is x.

00:13 I know that the dimes she has is going to be twice as much, so two times x.

00:18 She has five more nickels than dimes.

00:22 And so let me just kind of keep track.

00:24 X is the quarters.

00:26 I'll just abbreviate it, core.

00:29 And so this is the dimes, five more nickels than dimes.

00:39 So in other words, 2x plus five is the nickels.

00:45 So i'll just put nick.

00:48 How much of each coin does she have? well, now we have to be careful because you can't just add these together.

00:54 You also have to add their value.

00:56 So i'm going to put 25 multiplied by x because that's how much each of those quarters is worth.

01:02 I'm going to put a 10 multiplied by the 2x because that's how much each dime is worth.

01:08 And i'm going to put a 5 multiplied by all that 2x plus 5 for the nickels because each nickels obviously 5 cents.

01:16 And say this whole thing is equal to, i use whole numbers, didn't i? so i should say 300 cents.

01:24 Hopefully that makes sense because instead of otherwise i'd have to put 0 .25, 0 .10.

01:29 Let's just keep with whole numbers here and say 3.

01:32 Dollars is 300 cents.

01:35 All right, let's do some simplifying...

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