Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To solve for the fraction, let's use variables to represent its numerator and denominator. Let's denote the numerator by [tex]\( n \)[/tex] and the denominator by [tex]\( d \)[/tex].
We are given two conditions:
1. When 1 is subtracted from the numerator, the fraction becomes [tex]\( \frac{1}{3} \)[/tex].
2. When 8 is added to the denominator, the fraction becomes [tex]\( \frac{1}{4} \)[/tex].
### Step 1: Set Up the Equations from the Conditions
First condition: When 1 is subtracted from the numerator:
[tex]\[ \frac{n - 1}{d} = \frac{1}{3} \][/tex]
Cross-multiplying to eliminate the fraction:
[tex]\[ 3(n - 1) = d \quad \Rightarrow \quad 3n - 3 = d \quad \Rightarrow \quad d = 3n - 3 \quad \text{(Equation 1)} \][/tex]
Second condition: When 8 is added to the denominator:
[tex]\[ \frac{n}{d + 8} = \frac{1}{4} \][/tex]
Cross-multiplying to eliminate the fraction:
[tex]\[ 4n = d + 8 \quad \Rightarrow \quad d = 4n - 8 \quad \text{(Equation 2)} \][/tex]
### Step 2: Solve the System of Equations
Now, we have two equations:
1. [tex]\( d = 3n - 3 \)[/tex]
2. [tex]\( d = 4n - 8 \)[/tex]
Since both expressions equal [tex]\( d \)[/tex], we can set them equal to each other:
[tex]\[ 3n - 3 = 4n - 8 \][/tex]
Isolate [tex]\( n \)[/tex]:
[tex]\[ 3n - 4n = -8 + 3 \quad \Rightarrow \quad -n = -5 \quad \Rightarrow \quad n = 5 \][/tex]
Substitute [tex]\( n = 5 \)[/tex] back into one of the original equations to find [tex]\( d \)[/tex]. We can use either equation, but let’s use Equation 1:
[tex]\[ d = 3n - 3 \quad \Rightarrow \quad d = 3(5) - 3 \quad \Rightarrow \quad d = 15 - 3 \quad \Rightarrow \quad d = 12 \][/tex]
### Step 3: Verify the Solution
The fraction is [tex]\( \frac{n}{d} = \frac{5}{12} \)[/tex].
Check the first condition:
[tex]\[ \frac{n - 1}{d} = \frac{5 - 1}{12} = \frac{4}{12} = \frac{1}{3} \][/tex]
This is correct.
Check the second condition:
[tex]\[ \frac{n}{d + 8} = \frac{5}{12 + 8} = \frac{5}{20} = \frac{1}{4} \][/tex]
This is also correct.
### Final Answer
The fraction is:
[tex]\[ \frac{5}{12} \][/tex]
We are given two conditions:
1. When 1 is subtracted from the numerator, the fraction becomes [tex]\( \frac{1}{3} \)[/tex].
2. When 8 is added to the denominator, the fraction becomes [tex]\( \frac{1}{4} \)[/tex].
### Step 1: Set Up the Equations from the Conditions
First condition: When 1 is subtracted from the numerator:
[tex]\[ \frac{n - 1}{d} = \frac{1}{3} \][/tex]
Cross-multiplying to eliminate the fraction:
[tex]\[ 3(n - 1) = d \quad \Rightarrow \quad 3n - 3 = d \quad \Rightarrow \quad d = 3n - 3 \quad \text{(Equation 1)} \][/tex]
Second condition: When 8 is added to the denominator:
[tex]\[ \frac{n}{d + 8} = \frac{1}{4} \][/tex]
Cross-multiplying to eliminate the fraction:
[tex]\[ 4n = d + 8 \quad \Rightarrow \quad d = 4n - 8 \quad \text{(Equation 2)} \][/tex]
### Step 2: Solve the System of Equations
Now, we have two equations:
1. [tex]\( d = 3n - 3 \)[/tex]
2. [tex]\( d = 4n - 8 \)[/tex]
Since both expressions equal [tex]\( d \)[/tex], we can set them equal to each other:
[tex]\[ 3n - 3 = 4n - 8 \][/tex]
Isolate [tex]\( n \)[/tex]:
[tex]\[ 3n - 4n = -8 + 3 \quad \Rightarrow \quad -n = -5 \quad \Rightarrow \quad n = 5 \][/tex]
Substitute [tex]\( n = 5 \)[/tex] back into one of the original equations to find [tex]\( d \)[/tex]. We can use either equation, but let’s use Equation 1:
[tex]\[ d = 3n - 3 \quad \Rightarrow \quad d = 3(5) - 3 \quad \Rightarrow \quad d = 15 - 3 \quad \Rightarrow \quad d = 12 \][/tex]
### Step 3: Verify the Solution
The fraction is [tex]\( \frac{n}{d} = \frac{5}{12} \)[/tex].
Check the first condition:
[tex]\[ \frac{n - 1}{d} = \frac{5 - 1}{12} = \frac{4}{12} = \frac{1}{3} \][/tex]
This is correct.
Check the second condition:
[tex]\[ \frac{n}{d + 8} = \frac{5}{12 + 8} = \frac{5}{20} = \frac{1}{4} \][/tex]
This is also correct.
### Final Answer
The fraction is:
[tex]\[ \frac{5}{12} \][/tex]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.