Q47. - Algebra 2

Question

By Ohio law, when children are napping, the number of children per childcare staff member may be as many as twice the maximum listed at the right. Write and solve an inequality to determine how many staff members are required to be present in a room where 17 children are napping and the youngest child is 18 months old.

Step-by-Step Solution

Answer

The required inequality is $14x\ge 17$.

At least 2 staff members are required to be present in a room where 17 children are napping and the youngest child is 18 months old.

Step-1 – Use the definition of inequality

An inequality is a comparison between two values, showing if one is less than, greater than, or not equal to another value. 

Mathematical operators used are as follows:
 
 a ≠ b : a is not equal to b
 a < b : a is less than b
 a > b : a is greater than b
 a ≤ b : a is less than or equal to b
 a ≥ b : a is greater than or equal to b.

Step-2 &ndash; Analyze the information given

When children are napping, the number of children per childcare staff member may be as many as twice the maximum listed in the table below. 

Step-3 &ndash; Analyze the table

As per question we have to determine how many staff members are required to be present in a room where 17 children are napping and the youngest child is 18 months old.

Observe that in table there are many possible situations listed but since in our case youngest child is at least 18 months old we need to focus on the row that says:

“Every 7 toddlers who are at least 18 months old, but less than 30 months old”

Step-4 &ndash; Construct the inequality

Now since double the numbers of children are allowed per staff member when the children are sleeping one staff member can take care of $7\times 2=14$ toddlers above age of 18 months.

If we let x be the number of staff members that must be present, then we must have 

Maximum number of children x staff members can attend to is 14x. 

But since total number of children napping is 17, our inequality will be:

$14x\ge 17$

Step-5 &ndash; Solve the inequality

We have $14x\ge 17$ to find a solution we try to isolate x by itself. Note that 14 is being multiplied with ‘x’ so to isolate x we do the opposite operation that is divide both sides by 14 as shown: 

$\begin{array}{l}\frac{14x}{14}\ge \frac{17}{14}\\ x\ge 1.214\end{array}$

Now since number of staff members cannot be a fraction it has to be a whole number $x$   must at least be equal to 2.

Thus, At least 2 staff members are required to be present in a room where 17 children are napping and the youngest child is 18 months old.