Finding Three Numbers Given Conditions
We are given that the sum of three numbers is 120. The second number is eight less than the first, and the third number is 1 1/2 times the second.
1. Let the first number be x.
2. The second number is eight less than the first, so it is x – 8.
3. The third number is 1 1/2 times the second, so it is 1.5(x – 8).
4. The sum of the three numbers is 120: x + (x – 8) + 1.5(x – 8) = 120.
5. Combine like terms: 3.5x – 20 = 120.
6. Add 20 to both sides: 3.5x = 140.
7. Divide both sides by 3.5: x = 40.
8. Substitute x back into the expressions for the second and third numbers: Second number = 40 – 8 = 32, Third number = 1.5 * 32 = 48.
Conclusion: The first number is 40, the second number is 32, and the third number is 48.
Calculating -10 Decreased by the Difference Between 6 and -12
We are given the expression “-10 decreased by the difference between 6 and -12” and want to find its value.
1. Translate the expression into mathematical symbols: -10 – (6 – (-12)).
2. Simplify the expression inside the parentheses: 6 – (-12) = 6 + 12 = 18.
3. Substitute this value back into the original expression: -10 – 18.
4. Perform the subtraction: -10 – 18 = -28.
Conclusion: The value of “-10 decreased by the difference between 6 and -12” is -28.
Calculating 5 Times the Sum of c and d Increased by Twice the Sum of 3c and d
1. Translate the expression into mathematical symbols: The expression translates to 5(c + d) + 2(3c + d).
2. Break down the first part: 5(c + d) means you multiply 5 by the sum of c and d.
3. Distribute the 5 across (c + d): This results in 5c + 5d.
4. Break down the second part: 2(3c + d) means you multiply 2 by the sum of 3c and d.
5. Distribute the 2 across (3c + d): This results in 6c + 2d.
6. Combine the two results: 5c + 5d (from step 3) and 6c + 2d (from step 5) are added together.
7. Perform the addition: 5c + 5d + 6c + 2d = 11c + 7d.
Conclusion: The value of “5 times the sum of c and d increased by twice the sum of 3c and d” is 11c + 7d.
Finding the Coordinate of Point C on a Number Line
We are given points A at -2 2/3 and B at 1 1/4. We need to find the coordinate of point C such that C is twice as far from A as it is from B.
Step 1: Convert A and B to improper fractions.
A = -2 2/3 becomes A = -8/3
B = 1 1/4 becomes B = 5/4
Step 2: Define x as the distance from B to C.
We let x be the distance from B to C because we know that C is twice as far from A as it is from B.
Step 3: Calculate the distance from A to C.
The distance from A to C would be 2x because C is twice as far from A as it is from B.
Step 4: Write the equation for the coordinate of C.
The coordinate of C can be represented as B + x or A + 2x.
Step 5: Equate the two expressions for C and solve for x.
B + x = A + 2x
5/4 + x = -8/3 + 2x
x = 47/12
Step 6: Substitute x back into the equation for C.
C = B + x
C = 5/4 + 47/12
C = 31/6
Conclusion: The coordinate of point C is 31/6.