Question

To get a chance to win a car, you must guess the number of keys in a jar to within 5 of the actual number. Those who are within this range are given a key to try in the ignition of the car. Suppose there are 587 keys in the jar. Write and solve an equation to determine the highest and lowest guesses that will give contestants a chance to win the car.

Short answer

The highest and lowest guesses that will give contestants a chance to win the car are $\mathbf{592}$keys and 582 keys respectively.

Step-by-step solution

– Description of step.

Let $a$ be the guess number.

Total number of keys in the jar is 587.

Write an equation to determine the highest and lowest guesses that will give contestants a chance to win the car using the information in step 1.

$\left|a-587\right|=5$

– Description of step.

For any real numbers $p$ and $q$, where$q\ge 0$,

Case 1) if $\left|p\right|=q$ then $p=q$ or

Case 2) if $\left|p\right|=q$then $p=-q$

– Description of step.

Apply case 1 mentioned in step 2 to $\left|a-587\right|=5$

and simplify.

$\begin{array}{l}a-587=5\\ a-587+587=5+587\\ a=592\end{array}$

– Description of step.

Apply case 2 mentioned in step 2 to $\left|a-587\right|=5$and simplify.

$\begin{array}{l}a-587=-5\\ a-587+587=-5+587\\ a=582\end{array}$

Therefore, the highest and lowest guesses that will give contestants a chance to win the car are 592 keys and 582 keys respectively