1). Sum of three numbers is 45. One number is five more than the second number.It is also twice the third.Find the numbers.
2).Perimeter of quadrilateral is 52in. Longest side is is three times as long as the shortest side. The other two sides are equally long and are 5 in. longer than the shortest side. Find length of all four sides.
m + n + x = 45
x = n + 5 so n = (x-5)
x = 2 m so m = x/2
x/2 + x-5 + x = 45
2.5 x = 50
25 x = 500
x = 20
n = x-5 = 15
m = x/2 = 10
L = 3 s
a = b = s+5
perimeter = L + s + a + b = 52
3 s + s + s + 5 + s + 5 = 52
6 s + 10 = 52
6 s = 42
s = 7 etc
1) Let's represent the three numbers as follows:
- First number: x
- Second number: y
- Third number: z
From the given information:
- The sum of the three numbers is 45, so we can write the equation: x + y + z = 45 (Equation 1)
- One number is five more than the second number, so we can write the equation: x = y + 5 (Equation 2)
- One number is also twice the third, so we can write the equation: x = 2z (Equation 3)
To solve this system of equations, we can use substitution or elimination method. Let's use substitution method in this case.
Substitute Equation 2 into Equation 1:
(y + 5) + y + z = 45
Combine like terms:
2y + z + 5 = 45
Subtract 5 from both sides:
2y + z = 40 (Equation 4)
Substitute Equation 3 into Equation 4:
2y + 2z = 40
Divide both sides by 2:
y + z = 20 (Equation 5)
Substitute Equation 2 into Equation 5:
(y + 5) + z = 20
Combine like terms:
y + z + 5 = 20
Subtract 5 from both sides:
y + z = 15 (Equation 6)
Now we have a system of two equations:
Equation 5: y + z = 20
Equation 6: y + z = 15
Since both Equation 5 and Equation 6 have the same left side (y + z), but different right sides, this system has no solution. Therefore, there are no numbers that satisfy the given conditions.
2) Let's represent the lengths of the sides as follows:
- Shortest side: x
- Longest side: 3x
- Other two sides: x + 5
From the given information, the perimeter of the quadrilateral is 52 inches. So, we can write the equation:
x + (3x) + (x + 5) + (x + 5) = 52
Combine like terms:
6x + 10 = 52
Subtract 10 from both sides:
6x = 42
Divide both sides by 6:
x = 7
Now that we know the length of the shortest side is 7 inches, we can find the lengths of the other sides:
Longest side: 3x = 3 * 7 = 21 inches
Other two sides: x + 5 = 7 + 5 = 12 inches
Therefore, the lengths of the four sides are:
- Shortest side: 7 inches
- Longest side: 21 inches
- Other two sides: 12 inches
1) To solve the first problem, let's assign variables to each of the three numbers. Let's call the first number x, the second number y, and the third number z. So we have the following equations based on the given information:
x + y + z = 45 (Sum of three numbers is 45)
x = y + 5 (One number is five more than the second number)
x = 2z (The first number is twice the third number)
To solve this system of equations, we can use substitution.
Start by substituting the value of x from the second equation into the first equation:
(y + 5) + y + z = 45
Now, simplify the equation:
2y + z + 5 = 45
Next, substitute the value of x from the third equation into the second equation:
2z = y + 5
Rearrange this equation:
y = 2z - 5
Now, we can substitute this value of y into the previous equation:
2(2z - 5) + z + 5 = 45
Now, simplify and solve for z:
4z - 10 + z + 5 = 45
5z - 5 = 45
5z = 50
z = 10
Now that we have the value of z, we can substitute it back into the equation y = 2z - 5 to find y:
y = 2(10) - 5
y = 20 - 5
y = 15
Finally, substitute the values of y and z back into the equation x = y + 5 to find x:
x = 15 + 5
x = 20
Therefore, the three numbers are x = 20, y = 15, and z = 10.
2) To solve the second problem, let's call the shortest side of the quadrilateral x. Since the longest side is three times as long as the shortest side, we can say that the longest side is 3x.
The other two sides are equally long and are 5 inches longer than the shortest side. So we can say that these two sides are x + 5.
To find the perimeter, we can add up all four sides:
Perimeter = x + (x + 5) + (x + 5) + 3x
Given that the perimeter is 52 inches, we can set up the equation:
52 = x + (x + 5) + (x + 5) + 3x
Simplify the equation:
52 = 6x + 10
Subtract 10 from both sides:
42 = 6x
Divide both sides by 6:
x = 7
Now, we can substitute this value of x back into the equation to find the lengths of the other sides:
Longest side = 3x = 3 * 7 = 21
Other two sides = x + 5 = 7 + 5 = 12
Therefore, the lengths of the four sides are 7 inches, 12 inches, 12 inches, and 21 inches.