Question

1. tom put 18 gallons of mid - grade gas in his truck and filled up his empty five - gallon gas can with regular gas for his lawnmower at home. he spent $59.91. the following week, he put 14 gallons of mid - grade and topped off his five - gallon can with just one gallon of regular gas. if he paid $39.75 and the prices remained the same, find the price per gallon of mid - grade and regular gas.
2. a total of $12,000 was invested in two types of bonds. one pays 8% simple interest while the other pays 10.5%. last year, the annual interest earned on the two investments was $1,145. how much was invested at each rate?
3. a collection of dimes and quarters is worth $9.55. if the quarters were dimes and the dimes were quarters, the total value would be $7.60. find the number of each coin.

Explanation:

6.

Step1: Define variables

Let $x$ be the price per - gallon of mid - grade gas and $y$ be the price per - gallon of regular gas.
The first purchase gives the equation $18x + 5y=59.91$.
The second purchase gives the equation $14x + y=39.75$.

Step2: Solve the system of equations

From the second equation $y = 39.75−14x$.
Substitute $y$ into the first equation:
\[

$$\begin{align*} 18x+5(39.75 - 14x)&=59.91\\ 18x + 198.75-70x&=59.91\\ 18x-70x&=59.91 - 198.75\\ - 52x&=-138.84\\ x&=\frac{-138.84}{-52}\\ x& = 2.67 \end{align*}$$

\]

Step3: Find the value of $y$

Substitute $x = 2.67$ into $y = 39.75−14x$:
\[

$$\begin{align*} y&=39.75-14\times2.67\\ y&=39.75 - 37.38\\ y&=2.37 \end{align*}$$

\]

Step1: Define variables

Let $x$ be the amount invested at $8\%$ and $y$ be the amount invested at $10.5\%$.
We have the system of equations:

$$\begin{cases}x + y=12000\\0.08x+0.105y = 1145\end{cases}$$

From the first equation $x = 12000 - y$.

Step2: Substitute and solve

Substitute $x = 12000 - y$ into the second equation:
\[

$$\begin{align*} 0.08(12000 - y)+0.105y&=1145\\ 960-0.08y+0.105y&=1145\\ -0.08y + 0.105y&=1145 - 960\\ 0.025y&=185\\ y&=\frac{185}{0.025}\\ y& = 7400 \end{align*}$$

\]

Step3: Find the value of $x$

Substitute $y = 7400$ into $x = 12000 - y$:
$x=12000 - 7400=4600$

Step1: Define variables

Let $d$ be the number of dimes and $q$ be the number of quarters.
We know that $0.1d+0.25q = 9.55$ (original value) and $0.25d + 0.1q=7.60$ (swapped - value).
Multiply the first equation by $4$ to get $0.4d + q=38.2$, so $q = 38.2-0.4d$.
Multiply the second equation by $10$ to get $2.5d+q = 76$.

Step2: Substitute and solve

Substitute $q = 38.2-0.4d$ into $2.5d + q=76$:
\[

$$\begin{align*} 2.5d+(38.2-0.4d)&=76\\ 2.5d-0.4d&=76 - 38.2\\ 2.1d&=37.8\\ d&=\frac{37.8}{2.1}\\ d& = 18 \end{align*}$$

\]

Step3: Find the value of $q$

Substitute $d = 18$ into $q = 38.2-0.4d$:
$q=38.2-0.4\times18=38.2 - 7.2 = 31$

Answer:

The price per gallon of mid - grade gas is $\$2.67$ and the price per gallon of regular gas is $\$2.37$.

7.