The denominator of a fraction is 3 more than twice the numerator. If both the numerator and denominator are decreased by 7 and the simplified result is 7/19 find the original fraction
original:
numerator: x
denominator: 2x+3
so the original fraction is
x/(2x+3)
new fraction:
(x-7)/(2x+3-7) = 7/19
19x - 133 = 14x - 28
5x = 105
x = 21
So you had 21/45
The denominator of a fraction is 3 more than twice the numerator.If both the numerator and the denominator of the fraction are increased by 5,the resulting fraction is 2/7.Find the fraction.
Oh, fractions, I have a love-hate relationship with those! Let's see if we can solve this riddle together.
We have two parts to tackle here: the numerator and the denominator.
Let's call the original numerator "x." According to the problem, the denominator is 3 more than twice the numerator, which translates to 2x + 3.
When both the numerator and denominator are decreased by 7, the new fraction becomes (x - 7)/(2x + 3 - 7). Simplifying this fraction, we get (x - 7)/(2x - 4).
We're given that the simplified result is 7/19, so we can set up an equation:
(x - 7)/(2x - 4) = 7/19
Now, I could go all mathsy on you and start solving this equation step by step, but let me introduce a little twist.
Why did the math book look sad? Because it had too many problems to solve!
Okay, let's focus! Multiplying both sides of the equation by (2x - 4) and cross-multiplying, we have:
19(x - 7) = 7(2x - 4)
Expanding that out gives us:
19x - 133 = 14x - 28
Now let's subtract 14x from both sides and add 133 to both sides:
5x = 105
Dividing both sides by 5 gives us:
x = 21
So, the original numerator is 21. Now, let's find the denominator!
The denominator is 3 more than twice the original numerator, which is 2(21) + 3 = 45.
Therefore, the original fraction is 21/45. But, hey, we can't leave it like that! We can simplify it further.
Why did the math book go to therapy? Because it had too many fractions to simplify!
Dividing both the numerator and denominator by their greatest common divisor, we find the simplified fraction: 7/15. Voila!
Let's break down the problem step-by-step:
Step 1: Let's assume the original fraction is \( \frac{n}{d} \), where \( n \) is the numerator and \( d \) is the denominator.
Step 2: According to the problem, the denominator of the fraction is 3 more than twice the numerator, so we can write the equation: \( d = 2n + 3 \).
Step 3: Now we are told that when both the numerator and denominator are decreased by 7, the simplified result is \( \frac{7}{19} \).
Step 4: So, we subtract 7 from both the numerator and the denominator to get the new fraction: \( \frac{n-7}{d-7} \).
Step 5: We can set up the equation using the fraction we obtained in step 4: \( \frac{n-7}{d-7} = \frac{7}{19} \).
Step 6: Since the original fraction is unknown, we need to solve the equation from step 5 to find the value of \( n \).
Simplifying the equation: \( \frac{n-7}{d-7} = \frac{7}{19} \)
Cross-multiplying: \( 19(n-7) = 7(d-7) \)
Expanding: \( 19n - 133 = 7d - 49 \)
Step 7: We also know from step 2 that \( d = 2n + 3 \), so we can substitute this into the equation from step 6:
\( 19n - 133 = 7(2n + 3) - 49 \)
Simplifying further: \( 19n - 133 = 14n + 21 - 49 \)
Combining like terms: \( 19n - 133 = 14n - 28 \)
Step 8: Solving for \( n \):
\( 19n - 133 = 14n - 28 \)
Subtracting 14n from both sides: \( 19n - 14n - 133 = -28 \)
Combining like terms: \( 5n - 133 = -28 \)
Adding 133 to both sides: \( 5n = 105 \)
Dividing both sides by 5: \( n = 21 \)
Step 9: Now that we know the numerator is 21, we can substitute this value into the equation from step 2 to find the value of the denominator:
\( d = 2n + 3 = 2(21) + 3 = 42 + 3 = 45 \)
Step 10: Therefore, the original fraction is \( \frac{21}{45} \).
To find the original fraction, we need to break down the problem into steps. Let's start by defining the numerator and denominator of the original fraction.
Let's assume the numerator is represented by the variable 'x', and the denominator is represented by the variable 'y'.
We know that the denominator of the original fraction is 3 more than twice the numerator, so we can set up the equation:
y = 2x + 3
Next, we're given that both the numerator and denominator are decreased by 7 to get the simplified fraction. We can set up another equation based on this information:
(x - 7) / (y - 7) = 7/19
We can substitute the value of 'y' from the first equation into the second equation:
(x - 7) / ((2x + 3) - 7) = 7/19
Simplifying the denominator, we get:
(x - 7) / (2x - 4) = 7/19
Now, we can cross-multiply and solve for 'x':
19(x - 7) = 7(2x - 4)
19x - 133 = 14x - 28
19x - 14x = -28 + 133
5x = 105
x = 21
Now that we have the value of 'x', we can substitute it back into the first equation to find the value of 'y':
y = 2x + 3
y = 2(21) + 3
y = 42 + 3
y = 45
Therefore, the original fraction is 21/45.