1. David recently had a garage sale and has amassed a pile of change. How many of each coin (penny, nickel, dime, quarters) does he have if:
a.There is at least 5 pennies, 5 nickels, 5 dimes and 5 quarters
b.There is only a prime number of each coin
c.The total value is $5.75
d.There is a total of 40 coins
I need help figuring this out, please help.
p+5n+10d+25q = 575
p+n+d+q = 40
Now you know that there are
5,7,11,13,17,19,23 as the number of each coin.
Since there are no odd cents, you know that p=5.
Now you have
5n+10d+25q = 570
n+d+q = 35
or,
n+2d+5q = 114
n+d+q = 35
subtracting, we have
d+4q = 79
4q must be 20,28,44,52,68,76
That means d is 59,51,35,27,11,3
But we know that d<40 and d is prime, so d=3 or 11
If
d=3, q=19, p=5 so n=13
and the value is 575
If
d=11, q=17, p=5 so n=7
again, the value is 575
To solve these problems, we need to use a combination of logical reasoning and algebraic equations. Let's break down each question:
a. There is at least 5 pennies, 5 nickels, 5 dimes, and 5 quarters:
Since there are at least 5 of each coin, we can start with each coin having 5 as a minimum. Let's represent the number of pennies as P, nickels as N, dimes as D, and quarters as Q. We can write the equation: P ≥ 5, N ≥ 5, D ≥ 5, Q ≥ 5. However, we don't know the exact number of each coin.
b. There is only a prime number of each coin:
In this case, we need to find prime numbers for the number of each coin. Prime numbers are numbers that are divisible only by 1 and themselves. The prime numbers up to 10 are 2, 3, 5, and 7. Let's represent the number of pennies as P (a prime number), nickels as N (a prime number), dimes as D (a prime number), and quarters as Q (a prime number). We don't know the exact values yet.
c. The total value is $5.75:
To find the total value, we need to multiply the number of each coin by its respective value: pennies (0.01), nickels (0.05), dimes (0.10), and quarters (0.25). Let's represent the number of pennies as P, nickels as N, dimes as D, and quarters as Q. The equation will be: 0.01P + 0.05N + 0.10D + 0.25Q = 5.75. We need to solve this equation to find the values of P, N, D, and Q.
d. There are a total of 40 coins:
Let's represent the total number of coins as T. The equation will be: P + N + D + Q = 40. We need to solve this equation to find the values of P, N, D, and Q.
To find the actual values of P, N, D, and Q in each case, we need to use algebraic techniques such as substitution, elimination, or matrices to solve the system of equations.