Kiran has 27 nickels and quarters in his pocket, worth a total of $2.75. a. Write a system of equations to represent the relationships between the number of nickels n, the number of dimes d, and the dollar amount in this situation.
The correct answer and explanation is:
To solve this problem, let’s define the variables clearly and carefully write the system of equations.
Given:
- Kiran has 27 coins total made up of nickels (n) and quarters (q).
- The total value of the coins is $2.75, which is equivalent to 275 cents (to avoid working with decimals).
- Each nickel is worth 5 cents, and each quarter is worth 25 cents.
a. System of Equations:
Let:
- nn = number of nickels
- qq = number of quarters
Equation 1: Total number of coins n+q=27n + q = 27
Equation 2: Total value of the coins (in cents) 5n+25q=2755n + 25q = 275
Explanation (300 words):
To solve problems involving combinations of different types of coins, it’s helpful to define variables for each coin type and form equations based on the total number of coins and their total value.
In this case, Kiran has a total of 27 coins, made up of nickels and quarters. So, the first equation simply reflects that the number of nickels plus the number of quarters equals 27.
Next, we look at the value of the coins. A nickel is worth 5 cents, and a quarter is worth 25 cents. If Kiran has nn nickels, their total value is 5n5n cents. Similarly, qq quarters would be worth 25q25q cents. Since the total amount is $2.75, we convert this into cents to avoid dealing with decimals, getting 275 cents. Thus, the second equation, 5n+25q=2755n + 25q = 275, expresses the total value of all the coins in terms of the two variables.
This system of equations can be solved using substitution or elimination. For example, from the first equation n=27−qn = 27 – q, and substituting into the second gives: 5(27−q)+25q=275⇒135−5q+25q=275⇒20q=140⇒q=75(27 – q) + 25q = 275 \Rightarrow 135 – 5q + 25q = 275 \Rightarrow 20q = 140 \Rightarrow q = 7
Then n=27−7=20n = 27 – 7 = 20. So, Kiran has 20 nickels and 7 quarters.
This approach illustrates how systems of equations help solve real-world money problems effectively.