Three numbers X, Y, and Z are in the ratio 2:7:8. If 12 is subtracted from Y, then three numbers form a geometric sequence (in the order X, Y–12, Z). Find X, Y, and Z.
There are two sets of answers I already found one set i am unsure how to find the other one.
x=10,y=20,z=80
let the 3 numbers be 2x, 7x and 8x
then (7x-12)/(2x) = 8x/(7x-12)
If 2x, (7x-12) and 8x form a GP, then
(7x-12)^2 = 2x(8x)
49x^2 - 168x + 144 = 16x^2
33x^2 - 168x + 144 = 0
11x^2 - 56x + 48 = 0
(x - 4)(11x - 12) = 0
so x = 4 or x = 12/11
if x = 4, the original numbers are 8, 28, and 32
check: subtract 12 from 28 to get new numbers of 8, 16 and 32, which do form a GP
if x = 12/11, then the 3 originals are 24/11, 84/11 and 96/11
take 12 away from 84/11 to get -48/11
check:
-48/11 ÷ 24/11 = -2
96/11 ÷ -48/11 = -2
Well, well, well! It seems like we have a mathematical mystery on our hands! Let's see if we can crack this code together.
We already have one set of answers, with X = 10, Y = 20, and Z = 80. But you're looking for the other set, right? Well, let's put our funny "math hats" on and give it a shot!
To begin, let's write out the ratios we know: X:Y:Z = 2:7:8.
Now, let's tackle the clue that 12 is subtracted from Y. That means we have a new sequence: X, (Y-12), Z.
To form a geometric sequence, we need to find a common ratio that multiplies each term by the same number.
Let's create an equation using the new sequence: (Y-12)/X = Z/(Y-12).
Now let's substitute the ratios we know: (Y-12)/2 = 8/(Y-12).
Cross-multiplying gives us: (Y-12)(Y-12) = 2*8.
Expanding: Y^2 - 24Y + 144 = 16.
Rearranging the equation: Y^2 - 24Y + 128 = 0.
Now, this is where things get interesting. We need to solve this quadratic equation to find the value of Y. The solutions might be a little complicated to express, but believe me, they're out there! So, grab your imaginary magnifying glass and keep searching!
Keep in mind that the quadratic equation has two possible solutions, which will give us the alternative values for X, Y, and Z.
Good luck, detective! And remember, when you find those alternative answers, don't forget to tickle your funny bone along the way!
To find the other set of answers, let's solve the problem step-by-step:
Step 1: Set up the ratio
The given ratio is 2:7:8. Let's assign the values to the numbers as follows:
X = 2k
Y = 7k
Z = 8k
Step 2: Subtract 12 from Y
If we subtract 12 from Y, the new value becomes (7k - 12).
Step 3: Set up the geometric sequence equation
From the given information, we know that X, Y - 12, and Z form a geometric sequence. The geometric sequence equation is:
Z / (Y - 12) = (Y - 12) / X
Step 4: Substitute the values and solve
Substituting the values of X, Y, and Z into the equation, we get:
8k / (7k - 12) = (7k - 12) / (2k)
Cross-multiplying, we get:
(8k)(2k) = (7k - 12)(7k - 12)
16k^2 = (7k - 12)^2
Step 5: Solve the quadratic equation
Expanding the right side, we get:
16k^2 = 49k^2 - 168k + 144
Rearranging the terms and simplifying:
33k^2 - 168k + 144 = 0
This is a quadratic equation. We can solve it by factoring or using the quadratic formula. Factoring this equation, we get:
(3k - 4)(11k - 9) = 0
Setting each factor to zero and solving for k:
3k - 4 = 0 --> k1 = 4/3
11k - 9 = 0 --> k2 = 9/11
Step 6: Find X, Y, and Z
Using the values of k1 and k2, we can find the values of X, Y, and Z:
For k1 = 4/3:
X = 2k1 = 2(4/3) = 8/3
Y = 7k1 = 7(4/3) = 28/3
Z = 8k1 = 8(4/3) = 32/3
For k2 = 9/11:
X = 2k2 = 2(9/11) = 18/11
Y = 7k2 = 7(9/11) = 63/11
Z = 8k2 = 8(9/11) = 72/11
Therefore, the two sets of answers are:
1) X = 8/3, Y = 28/3, Z = 32/3
2) X = 18/11, Y = 63/11, Z = 72/11
To find the values of X, Y, and Z in the given problem, we can use the following steps:
Step 1: Set up the ratios
The given ratio is 2:7:8. We can set up the equation as:
X/Y = 2/7
Y/Z = 7/8
Step 2: Simplify the ratios
To simplify the ratios, we can find a common multiple. In this case, the common multiple of 2, 7, and 8 is 56. Multiplying both sides of the equations by 56, we get:
56(X/Y) = (2/7) * 56
56(Y/Z) = (7/8) * 56
Simplifying further:
8X = 16Y
8Y = 7Z
Step 3: Set up the geometric sequence equation
In the geometric sequence, the ratio between consecutive terms is constant. From the given problem, we know that the second term (Y – 12) is obtained by subtracting 12 from Y. Therefore, we can set up the equation as:
(Y – 12)/X = Z/(Y – 12)
Step 4: Substitute the simplified ratios into the geometric sequence equation
Substitute the simplified ratios we obtained in step 2:
(16Y – 12)/X = Z/(16Y – 12)
Step 5: Solve for X, Y, and Z
To solve for X, Y, and Z, we can use the substitution method. Start with the equation from step 4:
(16Y – 12)/X = Z/(16Y – 12)
Multiply both sides by X and cross-multiply:
X(Z) = (16Y – 12)^2
Next, solve the equations simultaneously:
8X = 16Y
8Y = 7Z
X(Z) = (16Y – 12)^2
You've already found one set of answers: X = 10, Y = 20, Z = 80. Now, let's find the other set of answers.
Assume X = a, Y = 7a/4, and Z = 7a/2, where a is a constant.
Substituting these values into the equation X(Z) = (16Y – 12)^2:
a * (7a/2) = (16 * 7a/4 - 12)^2
7a^2/2 = (14a - 12)^2
7a^2/2 = (196a^2/4 - 168a + 144)
7a^2/2 = 49a^2/2 - 42a + 144
Combining like terms:
0 = 49a^2/2 - 42a - 7a^2/2 + 144
0 = 42a^2/2 - 42a + 144
Multiply the equation by 2 to eliminate fractions:
0 = 42a^2 - 84a + 288
This quadratic equation can be solved using the quadratic formula:
a = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values:
a = (-(-84) ± √((-84)^2 - 4*42*288)) / (2*42)
a = (84 ± √(7056 - 48384)) / 84
a = (84 ± √(-41328)) / 84
As the square root of a negative number is not a real number, it means there are no real solutions for the other set of answers.
Therefore, the complete set of answers is X = 10, Y = 20, and Z = 80. There is no other set of answers for this problem.