A jar contains pennies, nickels, and dimes. The number of nickels is 4 fewer than twice the number of pennies, . The number of dimes is 3 more than the number of nickels. The jar contains 47 coins in total.
Which equation must be true for the value of ?
A one-variable equation can count several related groups using the same letter. In a coin-count problem, write each kind of coin in terms of the given count, then add the counts to the stated total. For “fewer than twice,” double before subtracting. Build each new count from the group named in its comparison.
Hints
- Hint 1
A comparison tells you how to build one count from another. “4 fewer than twice” means multiply by first, then subtract . What expression counts the nickels?
- Hint 2
The dimes are compared with the nickels, not the pennies. Start with your whole nickel expression and add . Then combine the constant terms, the numbers without .
- Hint 3
A total includes every coin once. Add the penny count, the nickel count, and the dime count, then set their sum equal to .
Step-by-step
Count each kind of coin
Step 1Write the nickel count
No Desmos needed. You're choosing an equation, not solving one. Twice the pennies is . The nickels are fewer than that doubled count, so subtract afterward:
- Step 2
Build the dime count from nickels
The dimes are more than the nickels, so start with the entire nickel count and add :
Combine the constant terms:
- Step 3
Add all three kinds of coins
The -coin total includes pennies, nickels, and dimes, so add their counts:
That's the equation that must be true for the number of pennies, . Choice D.