1. To determine the number of deer in a game preserve, a conservationist catches 309 deer, tags them and lets them loose. Later,324 deer are caught ; 162 of them are tagged. How many deer are in the preserve?
2. q+9/3 + q-5/5 = 14/3
3. 6/v =9/v -1/11
4. x/x+7 -7/x-7 = x² +49/x² -49
5. y/10 – y/25=1/10
1. To determine the number of deer in the preserve, we can set up a proportion using the number of tagged deer and the total number of deer caught.
Let x be the total number of deer in the preserve.
So the proportion can be written as:
tagged deer / total deer = tagged deer caught / total deer caught
162 / x = 309 / 324 (since 162 deer were tagged out of 324 caught)
Now we can cross-multiply and solve for x:
162 * 324 = x * 309
52512 = 309x
x = 52512 / 309
x ≈ 170
Therefore, there are approximately 170 deer in the preserve.
2. To solve the equation q + 9/3 + q - 5/5 = 14/3, we can simplify it step by step.
First, combine like terms:
2q + 4/5 = 14/3
Then, multiply every term by the least common denominator (LCD), which is 15 in this case, to eliminate the denominators:
3 * (2q + 4/5) = 15 * (14/3)
6q + 12/5 = 70/3
Next, simplify the equation further by distributing and simplifying:
(6q * 5 + 12) / 5 = 70/3
(30q + 12) / 5 = 70/3
To get rid of the fraction, cross-multiply:
3(30q + 12) = 5 * 70
90q + 36 = 350
Now, isolate the variable q by subtracting 36 from both sides:
90q = 350 - 36
90q = 314
Finally, divide both sides of the equation by 90 to solve for q:
q = 314 / 90
q ≈ 3.49
Therefore, q is approximately equal to 3.49.
3. To solve the equation 6/v = 9/v - 1/11, we can simplify it step by step.
First, let's combine the fractions on the right side:
6/v = (9 - 1/11) / v
Next, find a common denominator for the fractions on the right side:
6/v = (99/11 - 1/11) / v
Now, simplify the right side:
6/v = 98/11 / v
To get rid of the fraction, cross-multiply:
11 * 6 = 98 * v
Simplify further:
66 = 98v
Now, isolate the variable v by dividing both sides of the equation by 98:
v = 66 / 98
v ≈ 0.673
Therefore, v is approximately equal to 0.673.
4. To solve the equation x/(x+7) - 7/(x-7) = x² + 49/x² - 49, we can simplify it step by step.
First, let's find a common denominator for the fraction terms:
(x(x-7))/((x+7)(x-7)) - 7(x+7)/((x+7)(x-7)) = x² + 49/x² - 49
Next, simplify and combine like terms:
x(x - 7) - 7(x + 7) = x² + 49 - 49(x+7)(x-7)
Expand and simplify further:
x² - 7x - 7x - 49 = x² + 49 - 49(x² - 49)
Simplify the equation:
x² - 14x - 49 = x² + 49 - 49x² + 49^2
Simplify and move all terms to one side:
0 = -x² - 14x - 49 + x² + 49x² - 49^2 - 49
Combine like terms:
0 = 50x² - 14x - 49^2 - 49
Simplify further:
0 = 50x² - 14x - 2401 - 49
Combine like terms:
0 = 50x² - 14x - 2450
To solve this quadratic equation, we can factor or use the quadratic formula.
After factoring or using the quadratic formula, we find that x has two possible values: x ≈ -2.87 and x ≈ 8.47.
Therefore, x is approximately equal to -2.87 or 8.47.
5. To solve the equation y/10 - y/25 = 1/10, we can simplify it step by step.
First, let's find a common denominator for the fraction terms:
(5y - 2y) / (50) = 1 / 10
Next, simplify and combine like terms:
3y / 50 = 1 / 10
Cross-multiply to eliminate the fractions:
10 * 3y = 50 * 1
Simplify further:
30y = 50
Now, isolate the variable y by dividing both sides of the equation by 30:
y = 50 / 30
y ≈ 1.67
Therefore, y is approximately equal to 1.67.
1. To determine the number of deer in the preserve, we can set up a proportion using the ratio of tagged deer to the total number of deer in the preserve.
Let's denote the total number of deer in the preserve as "x".
The proportion can be set up as:
(Number of tagged deer) / (Total number of deer) = (Number of tagged deer in the second catch) / (Total number of deer in the second catch)
Plugging in the given values:
162 / x = 324 / 309
To solve for x, we can cross-multiply:
162 * 309 = 324 * x
x = (162 * 309) / 324
So, the number of deer in the preserve is approximately 154.
2. To solve the equation q+9/3 + q-5/5 = 14/3, we can simplify the left side of the equation by finding a common denominator for the fractions.
The common denominator for 3 and 5 is 15, so we can rewrite the equation as:
(5*(q+9) + 3*(q-5))/15 = 14/3
Multiplying through by 15 to clear the denominators:
5*(q+9) + 3*(q-5) = 14 * 15/3
Simplifying the equation:
5q + 45 + 3q - 15 = 70
Combining like terms:
8q + 30 = 70
Subtracting 30 from both sides:
8q = 40
Dividing both sides by 8:
q = 5
So the value of q that satisfies the equation is 5.
3. To solve the equation 6/v = 9/v - 1/11, we can start by finding a common denominator for the fractions.
The common denominator between v and 11v (to accommodate the 1/11 fraction) is 11v.
Multiplying through by 11v to clear the denominators:
11v*(6/v) = 11v*(9/v) - 11v*(1/11)
Simplifying the equation:
66 = 99 - v
Combining like terms:
v = 99 - 66
v = 33
Therefore, the value of v that satisfies the equation is 33.
4. To solve the equation x/(x+7) - 7/(x-7) = x² + 49/x² - 49, we can start by finding a common denominator for the fractions.
The common denominator between (x+7) and (x-7) is (x+7)(x-7).
Multiplying through by (x+7)(x-7) to clear the denominators:
[(x+7)(x-7)] * [x/(x+7)] - [(x+7)(x-7)] * [7/(x-7)] = [(x+7)(x-7)] * [x² + 49/x² - 49]
Simplifying the equation:
x(x-7) - 7(x+7) = (x+7)(x-7)(x² + 49/x² - 49)
Expanding and simplifying both sides:
x² - 7x - 7x - 49 = (x² - 49)(x + 49/x² - 1)
Combining like terms on the left side:
x² - 14x - 49 = (x² - 49)(x + 49/x² - 1)
Expanding the right side:
x² - 14x - 49 = x³ + 49x - x² - 49x - 49 + x² - 49/x² + 49
Simplifying the equation:
x² - 14x - 49 = x³ + 49x - x² - 49x - 49 + x² - 49/x² + 49
Combining like terms:
0 = x³ - 14x + 49 - 49/x²
Multiplying through by x² to eliminate the fraction:
0 = x³*x² - 14x*x² + 49x² - 49
Simplifying:
0 = x^5 - 14x³ + 49x² - 49
At this point, solving for x analytically becomes challenging and may require numerical methods.