Question

If a letter is selected at random from the letters of the word TRIANGLE, then what is the probability that it will be a consonant? (1) \(2 / 7\) (2) \(3 / 8\) (3) \(5 / 8\) (4) \(1 / 8\)

Short answer

Solution: To find the probability, we can count the total number of letters in the word TRIANGLE and the number of consonants present in the word. Then, we can calculate the probability of selecting a consonant by dividing the number of consonants by the total number of letters. Total number of letters: 8 Number of consonants: 5 (T, R, N, G, L) Probability of selecting a consonant = (Number of consonants) / (Total number of letters) = 5 / 8 Therefore, the probability of selecting a consonant from the word TRIANGLE is 5/8.

Step-by-step solution

Count the total number of letters

Count the number of letters in the word TRIANGLE. There are 8 letters.

Identify the consonants

Consonants in the word TRIANGLE are T, R, N, G, and L. There are 5 consonants.

Calculate the probability

The probability of selecting a consonant is the ratio of the number of consonants to the total number of letters. This can be represented as follows: \[ \frac{\text{Number of Consonants}}{\text{Total Number of Letters}} = \frac{5}{8} \] Therefore, the correct answer is (3) \(\frac{5}{8}\).

Key concepts

Consonants

When exploring word structures, such as counting letters, we often differentiate between vowels and consonants. In the word "TRIANGLE," let's look at how consonants are defined and identified. Consonants are all the letters in the alphabet that are not vowels (A, E, I, O, U). Thus, any letter that doesn't make a vowel sound falls into this category.
In the example word "TRIANGLE," by listing each letter, we determine the consonants to be

• T
• R
• N
• G
• L

That's a total of 5 consonants out of 8. Being able to correctly identify these non-vowel letters is essential, especially for calculating probabilities in problems like this one.

Random Selection

The concept of random selection is crucial in understanding how probabilities are calculated in scenarios like picking a letter from a word. Random selection means that each letter has an equal chance of being picked. Such a process ensures that every potential outcome is considered fair and unbiased.
Imagine you have a bag with all the letters of "TRIANGLE." If you close your eyes and pick one, any of the letters could be chosen with equal likelihood. That's what makes the selection random. Hence, if you aim to find the probability of picking a consonant, you assume that you don't favor any letter over another. This basic concept is fundamental for solving problems of probability involving random processes.

Mathematical Problem Solving

At its core, mathematical problem-solving involves breaking down a problem into manageable steps, much like the solution provided to this exercise. Let's walk through this process:

First, define what you know. In this problem, we're given the word "TRIANGLE," which contains 8 letters.
Second, identify what you're solving for, which is the probability that a randomly selected letter is a consonant. To do this, you need two pieces of information: the total number of letters and the number of consonants, both of which are counted straightforwardly.
Finally, solve using a probability formula:

• Probabilities are calculated as a fraction: the number of successful outcomes over the total number of possible outcomes.
• For our problem, successful outcomes are selecting a consonant, and we have already counted 5 consonants in "TRIANGLE."
• Thus, the probability is \[\frac{5 \text{ consonants}}{8 \text{ total letters}} = \frac{5}{8}\]

This formula and clear step-by-step method is typical in mathematical problem-solving, ensuring clarity and correctness in the solution.