Calculating The Rate Of Doing A Piece Of Work By Different Number Of Men

 

Men and Days Problems: Worker Time Rates

In math, when more people join a job, the work gets done faster (it takes fewer days). If fewer people do the job, it takes longer (more days). This is called inverse relationship.

---

The "Man-Days" Formula

The total amount of work needed to finish a job can be measured in "Man-Days" (Men × Days). Because the total work stays the same, we use this simple formula:

M₁ × D₁ = M₂ × D₂

• M₁ = Number of men at first
• D₁ = Number of days the first group takes
• M₂ = Number of men in the second group
• D₂ = Number of days the second group will take

---

Practice Questions and Solutions (Ascending Difficulty)

Question 1: Very Simple (Doubling the Workers)

If 5 men can complete a piece of work in 20 days, how long will it take 10 men to do the same work?

Solution:

• First Group: M₁ = 5 men, D₁ = 20 days
• Second Group: M₂ = 10 men, D₂ = ?
• Apply Formula: 5 × 20 = 10 × D₂
• Calculate: 100 = 10 × D₂
  D₂ = 100 ÷ 10 = 10 days

Answer: It will take 10 men 10 days.

Question 2: Simple (Reducing the Workers)

If 6 builders can build a wall in 4 days, how many days will it take just 2 builders to build the same wall?

Solution:

• First Group: M₁ = 6, D₁ = 4
• Second Group: M₂ = 2, D₂ = ?
• Apply Formula: 6 × 4 = 2 × D₂
• Calculate: 24 = 2 × D₂
  D₂ = 24 ÷ 2 = 12 days

Answer: It will take 2 builders 12 days.

Question 3: Medium (Finding Number of Men)

A group of 8 farmers can clear a farmland in 6 days. How many farmers are needed to clear the same farmland in exactly 4 days?

Solution:

• First Group: M₁ = 8, D₁ = 6
• Second Group: M₂ = ?, D₂ = 4
• Apply Formula: 8 × 6 = M₂ × 4
• Calculate: 48 = M₂ × 4
  M₂ = 48 ÷ 4 = 12 farmers

Answer: 12 farmers are needed.

Question 4: Hard (More Men Joining Later)

12 men are hired to dig a trench, and they can finish it in 5 days. If 3 more men join the team before they start, how many days will it take them all together?

Solution:

1. Find the new number of men: They started with 12 men, and 3 more joined. So, M₂ = 12 + 3 = 15 men.
2. Identify values: M₁ = 12, D₁ = 5, M₂ = 15, D₂ = ?
3. Apply Formula: 12 × 5 = 15 × D₂
4. Calculate: 60 = 15 × D₂
   D₂ = 60 ÷ 15 = 4 days

BlogThis!

Calculating The Rate of Doing Work

 

Understanding "Rate of Doing Work" in Primary School Math

In primary school math, the rate of doing work tells us how much of a task (like painting a wall or filling a tank) someone or something can complete in a single unit of time, such as 1 hour, 1 day, or 1 minute.

---

The Formula

To solve these problems, we use one simple formula:

Rate of Work = Total Work Done ÷ Time Taken

Important Rule: In math, a single complete job is always represented as 1 whole. For example, if a person takes 5 days to complete a job, their daily rate of work is 1/5 of that job.

---

Practice Questions and Solutions (Ascending Difficulty)

Question 1: Very Simple (Direct Rate)

John can paint a fence in 4 hours. What fraction of the fence can he paint in just 1 hour?

Solution:

• Total Work: 1 whole fence
• Total Time: 4 hours
• Calculation: Rate = 1/4

Answer: John can paint 1/4 of the fence in 1 hour.

Question 2: Simple (Finding Total Time)

A water pump can fill 1/5 of a swimming pool in 1 hour. How many hours will it take the pump to fill the entire pool?

Solution:

• Rate of Work: 1/5 of the pool per hour
• Total Work: 1 whole pool
• Calculation: Total Time = 1 ÷ (1/5) = 1 × 5 = 5 hours

Answer: It will take the pump 5 hours to fill the entire pool.

Question 3: Medium (Working Together)

Mary can clean a classroom in 3 hours, and Jane can clean the same classroom in 6 hours. If they work together, what fraction of the classroom will they clean in 1 hour?

Solution:

• Mary's 1-hour rate: 1/3 of the room
• Jane

How To Calculate Area Of Triangles

 

Mathematics is full of fascinating shapes, but one of the most fundamental shapes you will ever encounter is the triangle. Whether you are looking at the roof of a house, a slice of pizza, or a yield traffic sign, triangles are everywhere!

In this post, we will define what a triangle is, explore its four primary types with visual diagrams, and solve 5 simple practice questions perfect for primary school students.

---

What is a Triangle?

A triangle is a closed, three-sided geometric shape. It is formed by connecting three straight line segments. Every triangle has exactly:

• 3 sides (the straight edges)
• 3 vertices (the corner points where the sides meet)
• 3 internal angles (the space inside the corners)

Fun Fact: No matter how big or small a triangle is, the sum of its three internal angles always adds up to exactly 180°.

---

The Four Main Types of Triangles

Triangles are generally classified by the lengths of their sides or the sizes of their angles. Here is how the four main types look and differ from one another:

Equilateral
All sides equal

Isosceles
2 sides equal

Scalene
No sides equal

Right-Angled
One 90° angle

1. Equilateral Triangle

An equilateral triangle is a triangle where all three sides are equal in length. Because the sides are equal, all three internal angles are also equal, each measuring exactly 60°.

2. Isosceles Triangle

An isosceles triangle has two sides of equal length and one side that is different. The angles opposite to the equal sides are also equal to each other.

3. Scalene Triangle

A scalene triangle is a triangle where all three sides have different lengths. Consequently, all three internal angles have different measurements as well.

4. Right-Angled Triangle

A right-angled triangle (or right triangle) is a triangle that has one internal angle that measures exactly 90° (called a right angle). The side opposite the right angle is always the longest side, known as the hypotenuse.

---

How to Calculate the Area of a Triangle

To find out how much space is inside a triangle, you just need to know two measurements: the Base (the bottom side) and the Height (the straight vertical distance from the top point down to the base).

← ——— Base (B) ———→

The formula is:

Area = ½ × Base × Height

Or simply: Multiply the base by the height, and then divide the answer by 2.

---

5 Simple Questions and Solutions for Primary School

Question 1

Find the area of a triangle with a base of 6 cm and a height of 4 cm.

Solution:

• Base = 6 cm, Height = 4 cm
• Area = ½ × Base × Height
• Area = ½ × 6 × 4
• Area = ½ × 24
• Answer: Area = 12 cm²

Question 2

A triangular biscuit has a base of 8 cm and a height of 5 cm. What is its area?

Solution:

• Base = 8 cm, Height = 5 cm
• Area = ½ × 8 × 5
• Area = ½ × 40
• Answer: Area = 20 cm²

Question 3

A right-angled triangle has a base of 10 cm and a height of 3 cm. Calculate its area.

Solution:

• Base = 10 cm, Height = 3 cm
• Area = ½ × 10 × 3
• Area = 5 × 3
• Answer: Area = 15 cm²

Question 4

The base of a triangle is 12 cm and its height is 6 cm. Find the area.

Solution:

• Base = 12 cm, Height = 6 cm
• Area = ½ × 12 × 6
• Area = 6 × 6
• Answer: Area = 36 cm²

Question 5

A small triangular flag has a height of 7 cm and a base of 4 cm. What is the area of the flag?

Solution:

• Base = 4 cm, Height = 7 cm
• Area = ½ × 4 × 7
• Area = 2 × 7
• Answer: Area = 14 cm²

---

Parents and Teachers: Feel free to change the numbers in these questions to give your children more practice! Happy learning!

Conversion From Base Two To Base Ten

 

Converting Base Two to Base Ten

To convert a number from Base Two (Binary) back to Base Ten (Decimal), we multiply each digit by its positional power of 2 (starting from 2⁰ on the far right) and add the results together.

Question 1: Convert 110₂ to Base Ten

Solution:
• (1 × 2²) + (1 × 2¹) + (0 × 2⁰)
• (1 × 4) + (1 × 2) + (0 × 1)
• 4 + 2 + 0 = 6

Answer: 110₂ = 6₁₀

Question 2: Convert 1001₂ to Base Ten

Solution:
• (1 × 2³) + (0 × 2²) + (0 × 2¹) + (1 × 2⁰)
• (1 × 8) + (0 × 4) + (0 × 2) + (1 × 1)
• 8 + 0 + 0 + 1 = 9

Answer: 1001₂ = 9₁₀

Question 3: Convert 1100₂ to Base Ten

Solution:
• (1 × 2³) + (1 × 2²) + (0 × 2¹) + (0 × 2⁰)
• (1 × 8) + (1 × 4) + (0 × 2) + (0 × 1)
• 8 + 4 + 0 + 0 = 12

Answer: 1100₂ = 12₁₀

Question 4: Convert 1111₂ to Base Ten

Solution:
• (1 × 2³) + (1 × 2²) + (1 × 2¹) + (1 × 2⁰)
• (1 × 8) + (1 × 4) + (1 × 2) + (1 × 1)
• 8 + 4 + 2 + 1 = 15

Answer: 1111₂ = 15₁₀

Question 5: Convert 10100₂ to Base Ten

Solution:
• (1 × 2⁴) + (0 × 2³) + (1 × 2²) + (0 × 2¹) + (0 × 2⁰)
• (1 × 16) + (0 × 8) + (1 × 4) + (0 × 2) + (0 × 1)
• 16 + 0 + 4 + 0 + 0 = 20

Answer: 10100₂ = 20₁₀

Introduction To Base Two

 

Introduction to Number Bases (Base Two)

In our everyday lives, we use Base Ten (Decimal), which relies on ten digits (0 through 9). However, computers and digital systems don't understand ten digits—they operate on electricity, which is either on or off.

This is where Base Two (Binary) comes in. The binary system uses only two digits: 0 and 1.

• 0 represents "off" (low voltage).
• 1 represents "on" (high voltage).

How Base Two Works

Just like base ten uses powers of 10 (1, 10, 100, 1000), base two uses powers of 2. The place values double each time you move to the left:

24 (Sixteens)	23 (Eights)	22 (Fours)	21 (Twos)	20 (Ones)
16	8	4	2	1

To show a number is in a specific base, we write a small subscript. For example, 13₁₀ (thirteen in base ten) is written as 1101₂ in base two.

Converting Base Ten to Base Two

To convert a number from base ten to base two, we use the repeated division method. Divide the number by 2, write down the remainder, and repeat until you get 0. Then, read the remainders from the bottom to the top.

Question 1: Convert 6₁₀ to Base Two

Solution:
• 6 ÷ 2 = 3 remainder 0
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1

Answer: 6₁₀ = 110₂

Question 2: Convert 9₁₀ to Base Two

Solution:
• 9 ÷ 2 = 4 remainder 1
• 4 ÷ 2 = 2 remainder 0
• 2 ÷ 2 = 1 remainder 0
• 1 ÷ 2 = 0 remainder 1

Answer: 9₁₀ = 1001₂

Question 3: Convert 12₁₀ to Base Two

Solution:
• 12 ÷ 2 = 6 remainder 0
• 6 ÷ 2 = 3 remainder 0
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1

Answer: 12₁₀ = 1100₂

Question 4: Convert 15₁₀ to Base Two

Solution:
• 15 ÷ 2 = 7 remainder 1
• 7 ÷ 2 = 3 remainder 1
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1

Answer: 15₁₀ = 1111₂

Question 5: Convert 20₁₀ to Base Two

Solution:
• 20 ÷ 2 = 10 remainder 0
• 10 ÷ 2 = 5 remainder 0
• 5 ÷ 2 = 2 remainder 1
• 2 ÷ 2 = 1 remainder 0
• 1 ÷ 2 = 0 remainder 1

Answer: 20₁₀ = 10100₂

Permutations andcombinations

CLICK HERE

When we talk of permutations and combinations in everyday talk we often use the two terms interchangeably. In mathematics, however, each of them has very specific meanings, and this distinction often causes problems.

In brief, the permutation of a number of objects is the number of different ways they can be ordered; i.e. which is first, second, third, etc. If you wish to choose some objects from a larger number of objects, the way you position the chosen objects is also important.

Combinations, on the other hand, one does not consider the order in which objects were chosen or placed, just which objects were chosen.

We could summarize permutations and combinations (very simplistically) as

Permutations - position important (although choice may also be important)

Combinations - chosen important,

This may help you to remember which is which.

Pictures on a wall

Suppose you have to put some pictures on the wall, and suppose you only have two pictures: A and B.

You could hang them

Order 1 : A first,then  B

 or

Order 2:B  first,then A

Not much of a choice, but it leads on to the difference between permutations and combinations.

In English we use the word "combination" loosely, without thinking if the order of things is important.

In other words:

"My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad.

"The combination to the safe was 472". Now we do care about the order. "724" would not work, nor would "247". It has to be exactly 4-7-2.

So, in Mathematics we use more precise language:

If the order doesn't matter, it is a Combination.
If the order does matter it is a Permutation.

So, we should really call this a "Permutation Lock"!

In other words:

A Permutation is an ordered Combination.

THE IMPORTANT DIFFERENCE

As mentioned above, there is an important difference between permutations and combinations. In this case, for permutations the order of events is important: order 1 is different from order 2. For combinations, however, it does not matter which picture was hung first. In this example there are two permutations            (A, B ≠ B, A), but only one combination (A, B = B, A).

Examples

1. Permutations with replacement: Put ten (or whatever) numbers in a hat. Take one out, write down the number, put the number back in the hat, then do it again. If you do this 4 times, there are
10 x 10 x 10 x 10 = 10^4 = 10,000
different permutations. Order matters - 1234 is not the same permutation as 2134. For example, a telephone number or a lock combination (I suppose it should be called a permutation lock, then).

2,Permutations without replacement: Put ten (or whatever) numbers in a hat. Take one out, write down the number, throw the number away, then do it again. If you do this 4 times, there are
10 x 9 x 8 x 7 = 10! / (10-4)! = 5040
different permutations. Order matters - 1234 is not the same permuation as 2134 (and 2234 e.g. just cant occur). For example, there are
15 x 14 x 13 x 12 = 15! / (15-4)! = 32760
different ways you can sink the first four (of 15) numbered billiard balls.

3.Combinations without replacement: These are like permutations without replacement, except now order doesn't matter - 1234 and 2134 represent the same combination. Suppose you have 4 unique items, like these four digits. You can rearrange them in 4! ways - choose any of the 4 as the first digit, any of the remaining 3 as the second digit, and so on. (And so here is another example of a permutation without replacement!) Suppose you pick 6 of 40 numbers (without replacement) on a lottery ticket. There are 40! / (40-6)! = 93,963,542,400 permutations (!), but each permutation can be rearranged in 6! = 720 different ways, so there are only
[ 40! / (40-6)! ] / 6! = 130,504,920 combinations

If a lottery ticket costs $1, playing this lottery only makes sense from an economic perspective when the (accumulated) payout is greater than $130,504,920. If the payout was only half that, on average you'd lose 50 cents every time you played. However, many people don't see it that way. They figure that the whole dollar is lost as soon as they buy a ticket, but figure that it's well worth even the slimmest chance of winning $65 million (or even less) and buy the ticket anyway. Most governments take advantage of this discrepancy between price and value (or between exchange value and use value, if you're a Marxist), first by diverting 50% of lottery ticket sales to the state, second by taxing the winnings, usually at the highest personal income rate, and third by delivering the winnings as an annuity over 10-20 years instead of as an immediate cash payment. Together, these three measures can result in government retaining as much as 1 - 50% x 2/3 x 2/3 ~ 75% of ticket sales. (The actual number is closer to 50%; sellers of tickets keep 20-25% of the ticket price for their trouble.).

Keep learning!

Oyewole O Thomas

 

How to Solve Probability Problems on Cards

Probability is a branch of mathematics that deals with calculating the chance of a given event's occurrence, which is expressed as a number between 1 and 0. Or it is the numerical representation of an expectation.
An event with a probability of 1 can be considered a certainty: for example, the probability of a coin toss resulting in either "heads" or "tails" is 1, because there are no other options, assuming the coin lands flat.

 An event with a probability of 0.5 can be considered to have equal odds of occurring or not occurring: for example, the probability of a coin toss resulting in "heads" is 0.5, because the toss is equally as likely to result in "tails."

An event with a probability of 0 can be considered an impossibility: for example, the probability that the coin will land (flat) without either side facing up is 0, because either "heads" or "tails" must be facing up.

 A little paradoxical, probability theory applies precise calculations to quantify uncertain measures of random events.

Probability can be expressed mathematically as: the number of occurrences of a targeted event divided by the number of occurrences plus the number of failures of occurrences (this adds up to the total of possible outcomes):

p(x) = p(x)/[p(x) + p(y)]

CLICK HERE

Example
A card is drawn and replaced four times from a standard deck of 52 cards.
what is the probability of drawing a king

 a. exactly once?
b. exactly twice?
c. exactly three times?
d. exactly four times?
e. zero times?
f. Which is the most likely event(s) when drawing a card four times with replacement?

CLICK HERE

Solution:
 To figure out the answer you have to use binomial distribution.
The formula for exactly k successes in an n-trial experiment with replacement is:

(n choose k) * (probability of success)^k * (probability of failure)^(n-k)

Note that n = 4, since you have 4 trials.

Note also that the probability of success to draw a king is 1 in 13, or 1/13

A) The probability of drawing a king exactly once (k = 1) is:

(4 choose 1) * (1/13)^1 * (12/13)^3
= 4(0.0769)(0.787)
= 0.2421, or 24.2%

B) The probability of drawing a king exactly twice (k = 2) is:

(4 choose 2) * (1/13)^2 * (12/13)^2
= 6(0.0059)(0.852)
= 0.03, or 3%

C) The probability of drawing a king exactly three times (k = 3) is:

(4 choose 3) * (1/13)^3 * (12/13)^1
= 4(0.000455)(0.9231)
= .0017, or .17%

D) The probability of drawing a king exactly four times (k = 4) is:

(4 choose 4) * (1/13)^4 * (12/13)^0
= .0000035, round to 0.00004 or .004%

E) The probability of drawing a king exactly zero times (k = 0) is:

(4 choose 0) * (1/13)^0 * (12/13)^4
= 0.726, or 72.6%

F) The greatest probability is 72.6%, or never finding a king.

Note that if you were to sum all the probabilities found in part A through E, you would find that the probability is roughly equal to 1, or 100%, meaning that in your entire set you can only possibly draw a king zero, exactly one, two, three, or four times. This is a good way to check if you are correct.

Keep learning!
Oyewole O. Thomas