homework: quantity and value\nafter cleaning out your back pack, you find $1.70 in quarters,\ndimes, and…

homework: quantity and value\nafter cleaning out your back pack, you find $1.70 in quarters,\ndimes, and nickels at the bottom of the bag. there were\ntwice as many dimes as quarters. the number of nickels was\n2 less than 3 times the number of quarters.\nwrite an equation to determine the number of quarters, x.\nthen find the number of dimes and nickels

homework: quantity and value\nafter cleaning out your back pack, you find $1.70 in quarters,\ndimes, and nickels at the bottom of the bag. there were\ntwice as many dimes as quarters. the number of nickels was\n2 less than 3 times the number of quarters.\nwrite an equation to determine the number of quarters, x.\nthen find the number of dimes and nickels

Answer

Explanation:

Step1: Determine the value and quantity of each coin

  • A quarter is worth $0.25$, and the number of quarters is $x$.
  • A dime is worth $0.10$, and the number of dimes is $2x$ (since there are twice as many dimes as quarters).
  • A nickel is worth $0.05$, and the number of nickels is $3x - 2$ (since the number of nickels was 2 less than 3 times the number of quarters).

Step2: Write the equation for the total value

The total value of the coins is $1.70$. So, we can write the equation: $$0.25x+0.10\times(2x)+0.05\times(3x - 2)=1.70$$

Step3: Solve the equation

  • First, simplify the left - hand side of the equation: $$0.25x + 0.20x+0.15x-0.10 = 1.70$$
  • Combine like terms: $$(0.25 + 0.20+0.15)x-0.10=1.70$$ $$0.60x-0.10 = 1.70$$
  • Add $0.10$ to both sides: $$0.60x=1.70 + 0.10$$ $$0.60x=1.80$$
  • Divide both sides by $0.60$: $$x=\frac{1.80}{0.60}=3$$

Step4: Find the number of dimes and nickels

  • Number of dimes: $2x = 2\times3 = 6$
  • Number of nickels: $3x - 2=3\times3 - 2=9 - 2 = 7$

Answer:

The equation is $0.25x+0.10\times(2x)+0.05\times(3x - 2)=1.70$. The number of quarters $x = 3$, the number of dimes is $6$, and the number of nickels is $7$.