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Find out how working together changes things: 9 men and 12 boys need 12 days, while 12 men and 12 boys only need 10 days!
9 men and 12 boys finish a job in 12 days, 12 men and 12 boys finish it in 10 days. 10 men and 10 boys shall finish it in how many days?
10 men and 10 boys can finish the job in 12 days.
Let’s use the concept of work done by each individual in a day to solve this problem.
Step 1: Represent work and rates.
- Let the work required to complete the job be 1 unit.
- Let 1 man’s 1 day work be represented by m.
- Let 1 boy’s 1 day work be represented by b.
Step 2: Formulate equations based on given information.
- 9 men and 12 boys finish the job in 12 days: 9m + 12b = 1/12
- 12 men and 12 boys finish the job in 10 days: 12m + 12b = 1/10
Step 3: Solve for unknowns.
We want to find the number of days for 10 men and 10 boys to finish the job, which translates to 10m + 10b work units per day.
Method 1: Elimination
- Subtract equation 1 from equation 2: 3m = 1/30
- Solve for m: m = 1/90
- Substitute m back into equation 1: 9 * 1/90 + 12b = 1/12
- Simplify and solve for b: b = 1/180
- Calculate the work done by 10 men and 10 boys per day: 10 * 1/90 + 10 * 1/180 = 1/12
- This means they can finish the job in 12 days.
Method 2: Substitution
- Solve equation 1 for b: b = (1/12 – 9m) / 12
- Substitute this expression for b in equation 2: 12m + 12((1/12 – 9m) / 12) = 1/10
- Simplify and solve for m: m = 1/90
- Substitute m back into the expression for b from step 1: b = 1/180
- The rest of the steps are the same as method 1.
Therefore, 10 men and 10 boys can finish the job in 12 days.
Time, Work and Distance in Mathematics
Time, work, and distance are fundamental concepts in mathematics, often used in various real-life scenarios and problem-solving situations. Here’s a brief overview of each:
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Time: Time is a measure of duration or the interval between two events. It is typically measured in units such as seconds, minutes, hours, days, etc. Time is represented mathematically using numbers and can be added, subtracted, multiplied, and divided.
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Work: In mathematics, work is often used in the context of solving problems related to tasks that require effort or energy expenditure. It can represent various actions such as lifting objects, moving from one place to another, completing a project, etc. Work is often quantified in terms of units such as joules or foot-pounds.
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Distance: Distance is the measure of how far apart two objects or points are. It is a scalar quantity and is usually represented in units such as meters, kilometers, miles, etc. Distance can be measured along a straight path or a curved path and is a fundamental concept in geometry and physics.
When solving problems involving time, work, and distance, it’s common to use formulas and equations derived from these concepts. Some key formulas include:
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Distance = Speed × Time: This formula represents the basic relationship between distance, speed (or velocity), and time. It states that the distance traveled is equal to the product of the speed and the time taken to travel that distance.
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Average Speed = Total Distance ÷ Total Time: This formula is used to calculate the average speed when the total distance traveled and the total time taken are known.
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Work = Force × Distance: In physics, work is defined as the product of force and distance moved in the direction of the force. This formula represents the basic relationship between work, force, and distance.
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Work = Rate × Time: This formula is often used when considering work done at a constant rate over a period of time.
When solving problems involving time, work, and distance, it’s essential to clearly define the quantities involved and choose appropriate formulas or equations to solve for the unknowns. Additionally, units should be carefully considered and consistent throughout the problem-solving process.
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