Here is the problem:
A two-digit number is eight times the sum of its digits.When the number is added to the number obtained by reversing the digits,the sum is 99. Find the original number.
Please explain hot to solve it and it would help a lot to if you explain other algebra word problems like:
Distance rate and time problems,Mixture Problems,wind and water current problems,work problem,area problem,cost and value problems if you could explain or give a link to something helpful,
Thanks
To solve the given problem, let's break it down into steps:
Step 1: Understand the problem.
Read the problem carefully and identify what information is given and what needs to be found.
Step 2: Assign variables.
Assign variables to the unknowns in the problem. In this case, let's assume the tens digit of the two-digit number is 'x' and the units digit is 'y'.
Step 3: Set up equations.
Using the given information, set up equations to represent the problem. We know that the two-digit number is eight times the sum of its digits, so we can write the equation:
10x + y = 8(x + y)
We are also told that when the number is added to the number obtained by reversing its digits, the sum is 99. This can be written as:
10x + y + (10y + x) = 99
Step 4: Simplify and solve the equations.
Simplify the equations and solve them simultaneously.
From the first equation:
10x + y = 8x + 8y
2x = 7y (subtracting 8x and y from both sides)
From the second equation:
11x + 11y = 99
x + y = 9 (dividing both sides by 11)
We now have a system of two equations with two variables:
2x = 7y
x + y = 9
Using substitution or elimination methods, solve these equations to find the values of x and y.
By substituting the value of y from the second equation into the first equation:
2x = 7(9 - x)
2x = 63 - 7x
9x = 63
x = 63/9
x = 7
Substituting the value of x back into the second equation:
7 + y = 9
y = 2
Therefore, the original number is 72.
Now, let's move on to the other types of algebra word problems you mentioned:
Distance, Rate, and Time Problems:
These problems involve the formula distance = rate × time. You can solve them by identifying the given information (distance, rate, or time) and setting up the appropriate equation to find the missing value.
Mixture Problems:
In mixture problems, you have a mixture of two or more substances with known quantities and need to find the ratio or amount of each substance in the final mixture. These problems can be solved by setting up equations or using algebraic methods of solving systems of equations.
Wind and Water Current Problems:
Wind and water current problems involve calculating relative speeds or rates when moving in different directions. In these problems, you need to account for the direction and speed of the wind or water current and add or subtract it from the original speed to find the resultant speed.
Work Problems:
Work problems involve calculating the amount of work done by individuals or multiple individuals working together. You can use the formula work = rate × time or set up equations to solve these problems.
Area Problems:
Area problems involve calculating the area of shapes such as rectangles, triangles, circles, etc. You can use the appropriate formulas for each shape to find the area.
Cost and Value Problems:
Cost and value problems often require calculating the total cost or value of an item or set of items based on given prices or rates. You can set up equations to find these values or use proportions to calculate unknowns.
For more detailed explanations and practice problems on these algebra word problems, there are many online resources available. Websites like Khan Academy and MathisFun offer comprehensive lessons and examples for each type of problem.
To solve the two-digit number problem, let's break it down into steps:
1. Let's assume the two-digit number is represented by the variables "10x + y", where x represents the tens digit and y represents the units digit.
2. According to the problem, the number is eight times the sum of its digits. So, we can set up the equation: 10x + y = 8(x + y).
3. Next, we're given that when the number is added to the number obtained by reversing the digits, the sum is 99. In other words, (10x + y) + (10y + x) = 99.
4. Simplify the equation from step 3: 11x + 11y = 99.
5. Divide both sides of the equation by 11 to isolate x and y: x + y = 9.
6. Now we have a system of equations:
- 10x + y = 8(x + y) (equation from step 2)
- x + y = 9 (equation from step 5)
7. Solve the system of equations by substitution or elimination. Since equation 2 is already solved for x + y, we can substitute it into equation 1:
- 10x + y = 8(9).
Simplify: 10x + y = 72.
8. To eliminate the y term, subtract equation 2 from equation 1:
(10x + y) - (x + y) = 72 - 9.
Simplify: 10x - x = 63.
Simplify further: 9x = 63.
Divide both sides by 9: x = 7.
9. Substitute the value of x obtained in the previous step into equation 2 to find the value of y:
7 + y = 9.
Subtract 7 from both sides: y = 2.
10. The original two-digit number is 10x + y = 10(7) + 2 = 72.
Regarding your request for explanations on other algebra word problems, here are some helpful links for different types of problems:
- Distance, Rate, and Time Problems: https://www.mathsisfun.com/algebra/distance-rate-time.html
- Mixture Problems: https://www.mathplanet.com/education/algebra-1/age-word-problems-and-mixture-word-problems/mixture-word-problems
- Wind and Water Current Problems: https://www.qalaxia.com/#/ShareSession/share/7a5907d5-67d9-4f76-d582-5dd6db42c210/555f9e75-50f6-411f-8674-25dc2f05da71
- Work Problems: https://www.mathsisfun.com/algebra/work-word-problems.html
- Area Problems: https://www.qalaxia.com/#/ShareSession/share/7a5907d5-67d9-4f76-d582-5dd6db42c210/bf8cc3b4-072d-4f94-9c17-3540ab9377f2
- Cost and Value Problems: https://www.mathplanet.com/education/pre-algebra/ratios-and-proportions/investment-and-return-word-problems
These resources should help you understand and solve algebra word problems in various scenarios.