A string is \(48 cm\) long. It is cut into two pieces such that one piece is three times that of the other piece. What is the length of each piece?
Solution: Let us work on this problem using the IDEAL problem-solving method.
Step 1: Identify the problem.
We are given a length of a string, and we know that it is cut into two parts, whereby one part is three times longer than the other. As the length of the longer piece of string is dependent on the shorter string, we assume only one variable, say \(x\).
Step 2: Describe the outcome.
From the problem, we understand that we need to find the length of each piece of string. And we need the results such that the total length of both the pieces should be \(48 cm\).
Step 3: Explore possible strategies.
There are multiple ways to solve this problem. One way to solve it is by using the trial-and-error method. Also, as one length is dependent on another, the other way is to form an equation to solve for the unknown variable algebraically.
Step 4: Anticipate outcomes and act.
From the above step, we have two methods by which we can solve the given problem. Let's find out which method is more efficient and solve the problem by applying it.
Method 1
For the trial-and-error method, we need to assume value(s) one at a time for the variable and then solve for it individually until we get the total of 48.
That is, suppose we consider \(x=1\).
Then, by the condition, the second piece is three times the first piece.
\[\Rightarrow 3x=3(1)=3\]
Then the length of both pieces should be:
\[\Rightarrow 1+3=4\neq 48\]
Hence, our assumption is wrong. So, we need to consider another value. For this method, we continue this process until we find the total of \(48\). We can see that proceeding this way is time-consuming. So, let us apply the other method instead.
Method 2
In this method, we form an equation and solve it to obtain the unknown variable's value. We know that one piece is three times the other piece. Therefore, let the length of one piece be \(x\). Then the length of the other piece is \(3x\).
Now, as the string is \(48 cm\) long, it should be considered as a sum of both of its pieces.
\begin{align}&\Rightarrow x+3x=48 \\&\Rightarrow 4x=48 \\&\Rightarrow x=\frac{48}{4} \\&\Rightarrow x=12 \\\end{align}
So, the length of one piece is \(12cm\). The length of the other piece is \(3x=3(12)=36cm\).
Step 5: Look and learn
Let's take a look to see if our answers are correct. The unknown variable value we obtained is \(12\). Using it to find the other piece we get a value of \(36\). Now, adding both of them, we get:
\[\Rightarrow 12+36=48\].
Here, we got the correct total length. Hence, our calculations and applied method are right.
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