Can someone check my work for the following? I am supposed to use the five-step method to solve & check, but I don't know what that is. Anyway
1. One-third of a number is 5 less than half of the same number. What is the number?
My answer:
1/3x = 1/2x - 5
6(1/3x) = 6(1/2x) - 6( 5)
2x =3x - 30
2x - 2x = 3x - 30 - 2x
0 = x - 30
0 + 30 = x - 30 + 30
30 = x
x = 30
And to check:
1/3x = 1/2x - 5
1/3(30) = 1/2(30) - 5
30 / 3 = 30 / 2 - 5
10 = 15 - 5
10 = 10
2. The sum of a number and its reciprocal is 10/3. What is the number?
My Ans:
x+1/x= 10/3
(x^2+1)/x= 10/3
3 ( x^2+1)= 10 x
3x^2+3= 10x
3x^2-10x+3= 0
( 3x-1) ( x-3) = 0
3x-1= 0
x= 1/3
x-3= 0
x= 3
Check:
x+1/x= 10/3
3 + 1/3= 10/3
(9+1)/3= 10/3
10/3= 10/3
3. An investment of $1500 paid interest of $45 during a certain period. How much interest would an investment of $2000 invested at the same rate for the same length of time pay?
My Ans:
45/1,500=X/2,000
1,500X=45*2,000
1,500X=90,000
X=90,000/1,500
X=$60
Check:
45/1,500=60/2,000
90,000=90,000
4. In baseball a player's batting average is the ratio of hits to the number of times he or she has batted. Thus, if the player has an average of .3000, it means he or she got 300 hits in 1000 at bats. In order to have an average of .300, how many hits would the batter have to get in 400 times at bat?
My Ans:
batting average=(hits)/(times batted)
your batting average =.300
times batted=400
hits=x
.300 = x/400
(400).(.300)= 400x/400
120 = x
Check:
.300 = 120/400
.300 = 300
Thanks in advance everyone.
1, 2, and 4 are right. I'm not sure about 3.
#3 is correct
another way to do #3 is to simply set up a ratio
x/2000 = 45/1500
x = 45(2000)/1500
= 60
To solve the first problem, which is finding the number that satisfies the equation "One-third of a number is 5 less than half of the same number," you followed the correct steps:
1. Start with the equation: 1/3x = 1/2x - 5.
2. Multiply both sides of the equation by the least common denominator, in this case, 6.
3. Simplify the equation: 2x = 3x - 30.
4. Subtract 2x from both sides of the equation to isolate the variable: 0 = x - 30.
5. Add 30 to both sides to solve for x: 30 = x.
To check the solution:
1. Substitute the value of x (30) back into the original equation: 1/3(30) = 1/2(30) - 5.
2. Simplify the equation: 10 = 15 - 5.
3. Verify that both sides of the equation are equal: 10 = 10.
Therefore, your answer of x = 30 is correct.
For the second problem, you correctly followed the steps to solve for x using the equation "The sum of a number and its reciprocal is 10/3":
1. Start with the equation: x + 1/x = 10/3.
2. Multiply both sides of the equation by the least common denominator, in this case, 3x.
3. Simplify the equation: 3(x^2 + 1) = 10x.
4. Rearrange the equation to set it equal to zero: 3x^2 - 10x + 3 = 0.
5. Solve for x by factoring or using the quadratic formula: (3x - 1)(x - 3) = 0, which gives x = 1/3 or x = 3.
To check the solutions:
1. Substitute each value of x back into the original equation: x + 1/x = 10/3.
2. Verify that both sides of the equation are equal for each value of x: 10/3 = 10/3.
Therefore, your answers of x = 1/3 and x = 3 are correct.
For the third problem, where you need to find the amount of interest an investment of $2000 would earn at the same rate as an investment of $1500:
1. Start with the equation: 45/1500 = x/2000.
2. Multiply both sides of the equation by 2000.
3. Simplify the equation: 45 = 1500x.
4. Divide both sides of the equation by 1500 to solve for x: x = 45/1500 = 0.03.
To check the solution:
1. Substitute the value of x (0.03) back into the original equation: 45/1500 = x/2000.
2. Simplify the equation: 0.03 = 0.03.
Therefore, your answer of $60 for the interest earned on an investment of $2000 is correct.
For the fourth problem, where you need to find the number of hits necessary to achieve a batting average of .300 in 400 times at bat:
1. Start with the equation: x/400 = 0.3.
2. Multiply both sides of the equation by 400.
3. Simplify the equation: x = 120.
To check the solution:
1. Substitute the value of x (120) back into the original equation: 120/400 = 0.3.
2. Simplify the equation: 0.3 = 0.3.
Therefore, your answer of 120 hits is correct.
Great job on solving and checking these problems!
To solve the given problems, you can use the five-step method, which includes the following steps:
1. Read and understand the problem: Make sure you carefully read and comprehend the problem statement and what is being asked.
2. Assign variables: Identify the unknowns in the problem and assign variables to represent them. For example, in problem 1, you assigned the variable 'x' to represent the number.
3. Translate the problem into equations: Convert the problem statement into math equations using the assigned variables. This step involves understanding the relationship between the given information and the unknowns. For example, in problem 1, you translated the statement "One-third of a number is 5 less than half of the same number" into the equation "1/3x = 1/2x - 5".
4. Solve the equations: Use algebraic methods to solve the equations. You may need to simplify, combine like terms, isolate the variable, or use other techniques to solve for the unknowns.
5. Check your answer: Once you have obtained a solution, substitute the values back into the original problem to verify if the solution satisfies all the given conditions. This step ensures the answer is correct.
Now let's go through each of the problems and check your work:
Problem 1:
Your steps are correct. You assigned the variable 'x', translated the problem into an equation, solved it, and checked your answer. The final solution, x = 30, is correct. Well done!
Problem 2:
Again, your steps are correct. You assigned the variable 'x', translated the problem into an equation, solved it, and checked your answer. The solutions x = 1/3 and x = 3 are both correct answers since they satisfy the equation and make it equal to 10/3. Good job!
Problem 3:
Your work is correct. You assigned 'x' to the interest amount, translated the problem into an equation, solved it, and checked your answer. The solution, x = $60, is correct as it satisfies the equation and represents the interest amount for an investment of $2000. Great job!
Problem 4:
Once again, your steps are correct. You assigned 'x' to the number of hits, translated the problem into an equation, solved it, and checked your answer. The solution x = 120 is correct, as it represents the number of hits needed to maintain an average of .300 in 400 times at bat. Excellent!
Overall, your solutions are correct for all the problems. Keep up the good work and continue applying the five-step method to solve mathematical problems! If you have any further questions, feel free to ask.