Dudley has a number of 10c and 20c coins. If their total is $6, how many are there of each coin if there are:
a) equal numbers of each coin?
b) twice as many 10c coins as 20c coins
(a) 10x + 20x = 600
(b) 10*2x + 20x = 600
in each case, solve for x and in (b) find 2x
Let's solve the problem step by step:
a) Let's assume that Dudley has equal numbers of each coin.
Step 1: Set up the equation
Let's represent the number of 10c coins as x. The number of 20c coins will also be x, as both coins are equal in number. The total value can be expressed as the sum of the products of the number of coins and their respective values:
10c × x + 20c × x = $6
Step 2: Simplify the equation
Multiplying the coins by their respective values and combining like terms, we get:
10x + 20x = 6
Step 3: Combine like terms
Simplifying further, we have:
30x = 6
Step 4: Solve for x
Dividing both sides of the equation by 30, we get:
x = 6/30
Step 5: Calculate the number of coins
Since x represents the number of both 10c and 20c coins, there will be 6/30 = 1/5 of each coin.
Thus, Dudley has 1/5 of the total 10c and 20c coins.
b) Let's assume that Dudley has twice as many 10c coins as 20c coins.
Step 1: Set up the equation
Let's represent the number of 20c coins as x. Since there are twice as many 10c coins, we can represent their number as 2x. The total value equation remains the same:
10c × (2x) + 20c × x = $6
Step 2: Simplify the equation
Multiplying the coins by their respective values and combining like terms, we get:
20x + 20x = 6
Step 3: Combine like terms
Simplifying further, we have:
40x = 6
Step 4: Solve for x
Dividing both sides of the equation by 40, we get:
x = 6/40
Step 5: Calculate the number of coins
Since x represents the number of 20c coins, Dudley has 6/40 = 3/20 of these coins. As there are twice as many 10c coins, Dudley has 2 * (3/20) = 6/20 = 3/10 of the total 10c coins.
Therefore, in scenario (b), Dudley has 3/10 of the total 10c coins and 3/20 of the total 20c coins.
To solve this problem, we can set up a system of equations based on the given information.
Let's denote the number of 10c coins as "x" and the number of 20c coins as "y".
a) If there are equal numbers of each coin, we can set up the following equation:
0.10x + 0.20y = 6
Since we want an equal number of each coin, x = y.
Substituting y for x in the equation, we get:
0.10x + 0.20x = 6
0.30x = 6
x = 6 / 0.30
x = 20
Therefore, there are 20 coins of each denomination (10c and 20c).
b) If there are twice as many 10c coins as 20c coins, we can set up the following equation:
x = 2y
Substituting x = 2y in the equation 0.10x + 0.20y = 6, we get:
0.10(2y) + 0.20y = 6
0.20y + 0.20y = 6
0.40y = 6
y = 6 / 0.40
y = 15
Since x = 2y, substituting y = 15 in x = 2y, we get:
x = 2(15)
x = 30
Therefore, there are 30 coins of 10c denomination and 15 coins of 20c denomination.
To summarize:
a) If there are equal numbers of each coin, there are 20 coins of each denomination (10c and 20c).
b) If there are twice as many 10c coins as 20c coins, there are 30 coins of 10c denomination and 15 coins of 20c denomination.