Laverne has sixty-one coins, all of which are dimes and quarters. If the totoal value is $9.85, how many of each kind of coin she has?
Let's solve this problem step by step.
Step 1: Let's assume Laverne has x dimes and y quarters.
Step 2: We know that the total number of coins is 61. So we have the equation:
x + y = 61 (Equation 1)
Step 3: We also know that the total value of the coins is $9.85. The value of a dime is $0.10 and the value of a quarter is $0.25. So we can write another equation for the total value:
0.10x + 0.25y = 9.85 (Equation 2)
Step 4: Now we need to solve these two equations simultaneously.
Multiplying Equation 1 by 0.10, we get:
0.10x + 0.10y = 6.10 (Equation 3)
Subtracting Equation 3 from Equation 2, we eliminate the term with x:
0.10x + 0.25y - (0.10x + 0.10y) = 9.85 - 6.10
0.10x - 0.10x + 0.25y - 0.10y = 3.75
0.15y = 3.75
Step 5: Divide both sides of the equation by 0.15:
y = 3.75 / 0.15
y = 25
Step 6: Substitute the value of y back into Equation 1 to find x:
x + 25 = 61
x = 61 - 25
x = 36
Step 7: Laverne has 36 dimes and 25 quarters.
Therefore, Laverne has 36 dimes and 25 quarters.
To solve this problem, we can set up a system of equations. Let's assume Laverne has x number of dimes and y number of quarters.
The first equation represents the total number of coins:
x + y = 61
The second equation represents the total value of the coins:
0.10x + 0.25y = 9.85
Now we can solve this system of equations to find the values of x and y.
One way to solve this is by substitution. We can solve the first equation for x and substitute it into the second equation. Let's solve the first equation for x:
x = 61 - y
Now we substitute this value of x into the second equation:
0.10(61 - y) + 0.25y = 9.85
Simplifying the equation:
6.1 - 0.10y + 0.25y = 9.85
Combining like terms:
0.15y = 3.75
Dividing both sides by 0.15:
y = 3.75 / 0.15 = 25
Now we substitute this value of y back into the first equation to solve for x:
x + 25 = 61
x = 61 - 25 = 36
Therefore, Laverne has 36 dimes and 25 quarters.
Let D be the number of dimes and Q be the number of quarters. Use algebra.
D + Q = 61
10 D + 25 Q = 985
Solve that pair of equations.
10 D + 25 (61 -D) = 985
15 D = 540
D = 36
Q = 61 - 36 = 25