String p is 3 times as long as string q. string r is 3m longer than half the total length of strings p and q. If the total length of all three strings is 54oocm what is the length of string r.
String P is 3 times as long as string Q:
P = 3Q
String R is 3m longer than half the total length of strings P and Q:
R = (1/2)(P + 3Q) + 3
The total length of all three strings is 5400 cm:
P + Q + R = 5400
Substituting (such that everything is in terms of Q):
3Q + Q + (1/2)(3Q + 3Q) + 3 = 5400
4Q + (1/2)(6Q) + 3 = 5400
4Q + 3Q + 3 = 5400
7Q = 5400 - 3
7Q = 5397
Q = ?
After you solved for Q, get the values for P and R by substituting the value of Q to the equations we solved above.
Hope this helps~ `u`
1. String P is 3 times as long as string Q
so P=3Q
2. String R is 3m longer than half the total length of strings P and Q
so R=300+(3Q+Q)(1/2), or R=300+2Q
3. P+Q+R=5400
4. so solving for Q, 3Q+Q+2Q+300=5400
6. 6Q+300=5400
7. 6Q=5400-300
8. 6Q=5100
9. Q=850
10. P=850x3=2,550
11. R=((2,550+850)(1/2))+300=2,000
12. 850+2,550+2,000=5,400
To solve this problem, we can break it down into smaller steps:
Step 1: Define the variables
Let's define the lengths of the strings as follows:
- Length of string q: x
- Length of string p: 3x (since it is 3 times longer than q)
- Length of string r: 3 + (1/2)(3x + x) (3m longer than half the total length of strings p and q)
Step 2: Write the equation
The total length of all three strings is 540 cm, so:
x + 3x + 3 + (1/2)(3x + x) = 540
Step 3: Simplify the equation
Combine like terms:
4x + 3 + (4x/2) = 540
4x + 3 + 2x = 540
6x + 3 = 540
Step 4: Solve for x
Subtract 3 from both sides of the equation to isolate the variable:
6x + 3 - 3 = 540 - 3
6x = 537
Divide both sides of the equation by 6 to solve for x:
6x/6 = 537/6
x = 89.5
Step 5: Find the length of string r
Substitute the value of x back into the expression for the length of string r:
Length of string r = 3 + (1/2)(3x + x)
Length of string r = 3 + (1/2)(3 * 89.5 + 89.5)
Length of string r = 3 + (1/2)(268.5 + 89.5)
Length of string r = 3 + (1/2)(358)
Length of string r = 3 + 179
Length of string r = 182 cm
Therefore, the length of string r is 182 cm.