Question

Dave Bowers collects U.S. gold coins. He has a collection of 41 coins. Some are​ $10 coins, and the rest are​ $20 coins. If the face value of the coins is ​$590​, how many of each denomination does he​ have?Dave Bowers collects U.S. gold coins. He has a collection of 34 coins. Some are​ $10 coins, and the rest are​ $20 coins. If the face value of the coins is ​$500​, how many of each denomination does he​ have?

Answer 1

1. He has 23 coins in $10, and 18 coins in $20

2. He has 18 coins in $10, and 16 coins in $20

Step-by-step explanation:

The number of $10 and $20 are two variables that can only be determined simultaneously. The product of the number of each type of coin with the value of each coin gives the total amount owned.

Let the number of of $10 coins owned be x and that of $20 be y

If he has a collection of 41 coins then

x + y = 41

If the face value of the coins is ​$590

then,

10x + 20y = 590

solving both simultaneously

x + y = 41

x + 2y = 59

y = 59 - 41

= 18

x = 41- 18

= 23

It means he has 23 coins in $10, and 18 coins in $20.

If he has a collection of 34 coins then

x + y = 34

If the face value of the coins is ​$500

then,

10x + 20y = 500

solving both simultaneously

x + y = 34

x + 2y = 50

y = 50 - 34

= 16

x = 34- 16

= 18

It means he has 18 coins in $10, and 16 coins in $20

Answer 2

1. Dave has 23 ($10 coins) and 18 ($20) coins.

2. Dave has 18 ($10 coins) and 16 ($20) coins.

Step-by-step explanation:

1.

Let x be the number of $10 coins.

Then, the number of $20 coins will be 41-x.

The equation for the sum of money can be written as:

590 = 10x + 20 * (41-x)

590 = 10x + 820 - 20x

590 - 820 = -10x

-230 / -10 = x

x = 23

This means that Dave has 23 $10 coins and (41-23 = 18) 18 $20 coins that sum up to a face value of $590.

2.

Using the same priciple,

let x be the number of $10 coins

let 34-x be the number of $20 coins

Sum of money equation:

500 = 10x + 20 * (34-x)

500 = 10x + 680 - 20x

500 - 680 = -10x

-180 / -10 = x

x = 18

So, Dave has 18 $10 coins and (34-18 = 16) 16 $20 coins that add up to a face value of $500.