114. The candy box contains 36 orange and lemon candies. The probability of picking a lemon candy is [tex]\frac{1}{3}[/tex]. How many lemon candies do you need to add so that the probability of picking out a lemon candy is the same as the probability of picking out an orange candy?

A. 12
B. 15
C. 18
D. 24



Answer :

Sure! Let's break down the problem step-by-step:

1. Understanding the Problem:
- Total number of candies = 36
- Probability of picking a lemon candy [tex]\(\frac{1}{3}\)[/tex]

2. Determine the Initial Number of Lemon and Orange Candies:
- To find the number of lemon candies, we use the given probability:
[tex]\[ \frac{\text{Number of lemon candies}}{\text{Total number of candies}} = \frac{1}{3} \][/tex]
Given the total candies are 36, let's denote the number of lemon candies as [tex]\( L \)[/tex]:
[tex]\[ \frac{L}{36} = \frac{1}{3} \][/tex]
[tex]\[ L = 36 \times \frac{1}{3} = 12 \text{ lemon candies} \][/tex]
- Then, the number of orange candies is:
[tex]\[ 36 - 12 = 24 \text{ orange candies} \][/tex]

3. Set up the Equation to Find How Many Lemon Candies to Add:
- Let's denote the number of lemon candies to add as [tex]\( x \)[/tex].
- After adding [tex]\( x \)[/tex] lemon candies, the new number of lemon candies will be [tex]\( 12 + x \)[/tex].
- The total number of candies will now be [tex]\( 36 + x \)[/tex].
- We want the probabilities of picking a lemon candy and an orange candy to be equal, which means:
[tex]\[ \frac{12 + x}{36 + x} = \frac{24}{36 + x} \][/tex]
- At this point, we have two equal fractions with the same denominator:
[tex]\[ 12 + x = 24 \][/tex]

4. Solve for [tex]\( x \)[/tex]:
- To find [tex]\( x \)[/tex], simply solve the equation:
[tex]\[ 12 + x = 24 \][/tex]
[tex]\[ x = 24 - 12 \][/tex]
[tex]\[ x = 12 \][/tex]

Therefore, you need to add 12 lemon candies to make the probability of picking a lemon candy equal to the probability of picking an orange candy.

The correct answer is E. 12.