# Thread: solving equations in English

1. ## solving equations in English

Systems of equations can be easily solved through the matrix. But when English is used, the difficulty of the problem is compounded. I am stumped by this question, I cannot figure out how to solve this question. Please help.

Jack drives from her country home to the city, a distance of 380 km. He drives part of the way on back roads, at an average speed of 50 km/h, and part on highways, at an average speed of 100 km/h. The trip takes 5 hours. How far does he drive on each type of road?

Thanks.

2. Originally Posted by shenton
Systems of equations can be easily solved through the matrix. But when English is used, the difficulty of the problem is compounded. I am stumped by this question, I cannot figure out how to solve this question. Please help.

Jack drives from her country home to the city, a distance of 380 km. He drives part of the way on back roads, at an average speed of 50 km/h, and part on highways, at an average speed of 100 km/h. The trip takes 5 hours. How far does he drive on each type of road?

Thanks.
Translate this into an equation. Let $\displaystyle x \mbox{ km}$ be the distance travelled on back roads. Then $\displaystyle 380-x\mbox{ km}$ is the distance travelled on highways. Then the time taken is:

$\displaystyle t=x/50+(380-x)/100=5 \mbox{ hours}$

Now solve this equation for $\displaystyle x$.

RonL

3. Hello, shenton!

I use baby-talk with such word problems . . . it works for me.

Jack drives from her country home to the city, a distance of 380 km.
He drives part of the way on back roads, at an average speed of 50 km/h,
and part on highways, at an average speed of 100 km/h.
The trip takes 5 hours.
How far does he drive on each type of road?

There are a number of approaches to this problem.
If you prefer a system of equations, we can do it like this . . .

Let $\displaystyle x$ = number of km on back roads.
Let $\displaystyle y$ = number of km on highways.
. . We know that: .$\displaystyle x + y \:=\:380$ (1)

He drove $\displaystyle x$ km at 50 km/hr.
. . This took him $\displaystyle \frac{x}{50}$ hours.
He drove $\displaystyle y$ km at 100 km/hr.
. . This took him $\displaystyle \frac{y}{100}$ hours.
Since his total time was 5 hours, we have: .$\displaystyle \frac{x}{50} + \frac{y}{100} \:=\:5$
. . Multiply by 100: .$\displaystyle 2x + y \:=\:500$ (2)

Solve the system of equations: .$\displaystyle \begin{array}{cc}(1) \\ (2) \end{array}\begin{array}{cc} x + y & =\:380 \\ 2x + y & =\:500\end{array}$

Subtract (1) from (2): .$\displaystyle \boxed{x \,= \,120}$

Substitute into (1): .$\displaystyle 120 + y\:=\:380\quad\Rightarrow\quad\boxed{y \,=\,260}$

Therefore, Jack drove: .$\displaystyle \begin{array}{cc} 120\text{ km} & \text{on back roads} \\ 260\text{ km} & \text{on highways}\end{array}$

4. Thanks, Soroban for showing the 2 equations and the detailed workings.

This question and the quarters and dimes questions are one of the most difficult word problems.