A and B together can do a piece of work in 30 days. A having worked for 16 days, B finishes the remaining work alone in 44 days. In how many days shall B finish the whole work alone? 

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Join A and B on a 30-day project where A pitches in for 16 days. Then, watch as B takes over and wraps up the remaining work in 44 days. Curious about how many days B needs to finish the entire project solo? Let’s uncover the answer!

A and B together can do a piece of work in 30 days. A having worked for 16 days, B finishes the remaining work alone in 44 days. In how many days shall B finish the whole work alone?

A and B together can do the entire work in 30 days: a + b = 1​ / 30

A worked for 16 days, completing 16​/30 of the work.

The remaining work was finished by B alone in 44 days, which means B completed 14​/30 of the work in those 44 days.

We can set up two equations based on this information:

1. Combined work done: 16a + 44b = 14 / 30​ (since A did 16 days of work and B did the remaining 44 days)
2. Individual work rates: a+b = 1/30​ (as mentioned earlier)

Now we can solve for b, which represents the number of days it takes B to finish the whole work alone.

1. Solve the second equation for a: a = 1/30 ​− b
2. Substitute this expression for a in the first equation: 16((301​−b)) + 44b = 14​/ 30
3. Simplify and solve for b: 16​/30 −16b+44b = 14/30​ 28b = 2​/30 b = 1​/60

Therefore, it takes B 60 days to finish the whole work alone.

Linear Equations in Algebra

Linear equations are equations in which the highest power of any variable is 1. They are fundamental concepts in algebra and have numerous applications in various fields, including physics, economics, and computer science.

Linear equations in one variable:

• A linear equation in one variable can be written in the general form of ax + b = 0, where a and b are constants, and x is the unknown variable.
• The solution of the equation is the value of x that makes the equation true.
• There are various methods for solving linear equations in one variable, such as:
  • Inspection: If one side of the equation is already 0, the solution is the value of x on the other side.
  • Adding or subtracting terms: Add or subtract the same term to both sides of the equation to isolate x.
  • Multiplying or dividing by a number: Multiply or divide both sides of the equation by the same non-zero number to isolate x.

Example: Solve the equation 2x + 5 = 11.

Solution:

• Subtract 5 from both sides: 2x = 6
• Divide both sides by 2: x = 3

Linear equations in two variables:

• A linear equation in two variables can be written in the general form of ax + by = c, where a, b, and c are constants, and x and y are the unknown variables.
• The solution of the equation is the ordered pair (x, y) that makes the equation true.
• There are various methods for solving linear equations in two variables, such as:
  • Graphing: Graph the equation and find the point where the line intersects the x-axis and the y-axis.
  • Substitution: Solve one of the variables for its value in terms of the other variable and substitute it into the equation.
  • Elimination: Add or subtract the equations in a way that eliminates one of the variables, then solve for the remaining variable.

Example: Solve the system of equations:

x + y = 4 2x – y = 3

Solution:

• Add the two equations: 3x = 7
• Divide both sides by 3: x = 7/3
• Substitute 7/3 for x in the first equation: (7/3) + y = 4
• Solve for y: y = 5/3

Applications of linear equations:

• Linear equations can be used to model real-world situations, such as:
  • Motion: The distance traveled by an object can be modeled by the equation d = rt, where d is the distance, r is the rate, and t is the time.
  • Mixture problems: The concentration of a solution can be modeled by the equation c1v1 + c2v2 = cv, where c1 and c2 are the initial concentrations, v1 and v2 are the volumes of the solutions mixed, and c is the final concentration.
  • Supply and demand: The relationship between the price of a good and the quantity demanded can be modeled by the equation p = a – bq, where p is the price, q is the quantity demanded, and a and b are constants.

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