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# Tag Archives: logical reasoning

## Let’s Learn Mathematics (Primary Level) 3

Disclaimer: This Mathematics question is purely created for discussion purpose. Any resemblance to actual questions from books or schools is coincidental.

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We have discussed about simple algebra in Let’s Learn Mathematics (Primary Level). Though algebra is formally introduced in secondary schools, the model method is taught in primary schools as an introduction to algebra. Thus, you can see the importance of learning algebra.

The question of this post is considered advanced for primary students, nonetheless it is for primary students. Take your time to understand the question and the solution; they are helpful in learning algebra up to secondary schools.

Q: 3 shirts and a pair of trousers cost $63.50. A shirt and 2 pairs of trousers cost $27. Find the cost of a shirt.

A: First you need to draw the diagram to show the relationship of the equations, based on the question. The first sentence (3 shirts and a pair of trousers cost $63.50) forms an equation and the second equation forms another equation. [Please refer to The Almighty Algebra for notes.]

In the above diagram, the blue rectangle represents the shirt and the yellow triangle represents the pair of trousers.

If the student has been solving similar questions, he / she will start by adding or subtracting the two equations.

s = a shirt, t = a pair of trousers

Equation 1, 3s + t = 63.50

Equation 2, s + 2t = 27

Adding equation 1 and equation 2, 4s + 3t = 90.50

Subtracting equation 2 from equation 1, 2s − t = 36.50

This is where students may face the problem to continue solving the problem. They have all the equations but the equations are leading them to nowhere. No matter it is adding or subtracting, you end up with more equations with no solution.

You can apply any Mathematics operation to any equation.

The Mathematics operation includes adding, subtracting, multiplying and dividing. Yes, many students have not thought of multiplying the equations. Nonetheless, we do not encourage students to divide the equations because it will lead to decimals or fractions. It is difficult to solve the question, it is more difficult to solve the question with decimals or fractions.

Back to the question, the question is only asking for the cost of a shirt. So, to simplify the question, we must “eliminate” the cost of a pair of trousers.

Equation 1, 3s + t = 63.50

Equation 2, s + 2t = 27

The fastest way to get rid of “t” is to multiply 2 to equation 1 and then subtract equation 2 from the new equation (equation 3).

Equation 1 x2, 6s + 2t = 127 (Equation 3)

Equation 3 − Equation 2, 5s = 100 [There is no more “t” because 2t − 2t = 0]

s = 100 ÷ 5

s = 20

The cost of a shirt is $20.

## Let’s Learn Mathematics (Primary Level) 2

Disclaimer: This Mathematics question is purely created for discussion purpose. Any resemblance to actual questions from books or schools is coincidental.

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Q: In a seminar, the ratio of the number of men to the number of women was 2:3. On the following day, two more men attended the seminar. The ratio became 5:6. How many participants were there in the seminar at first?

Understanding the question

This is a question about ratio. The ratio has changed because the number of men has changed. Thus, steps to solve this question are:

1. Make the ratio with the same number of women on the first and second day of the seminar. It is because the number of women is unchanged.

2. Find out the change (difference) of the number of men on the first and second day of the seminar.

3. The difference of the number of men on the first and second day is 2, which is also equivalent to the answer from step 2.

4. Find the number of participants on the first day.

Step-by-step Answer

1. The men-women ratio on the first day => 2:3 = 4:6

The men-women ratio on the second day => 5:6

2. Comparing the men-women ratio on the first and second day => 4:6 and 5:6

The difference of the number of men on the first and second day = 5 – 4 = 1

3. 1 unit of difference = 2 more men attended the seminar on the second day.

4. Total units on the first day = 4 + 6 = 10

Total participants on the first day = 10 x 2

= 20 participants

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Checking the answer

On the first day, there were 20 participants with the ratio 2:3. Thus, the number of men on the first day was (20 x 2) ÷ 5 = 8.

On the second day, two more men attended the seminar. Thus, the number of men on the second day was 2 + 8 = 10.

The number of women on the first and second day was = 20 – 8 = 12.

The men-women ratio on the second day = 10:12

= 5:6

Thus, the answer, 20 participants, is correct.

Note to parents: If your child encounters difficult questions, asks him / her to check the answer. Checking the answer is a good way to reassure your child that he / she has answered the question correctly.

## Let’s Learn Mathematics (Primary Level) 1

Disclaimer: This Mathematics question is purely created for discussion purpose. Any resemblance to actual questions from books or schools is coincidental.

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Q: An orphanage owner fed some biscuits to the orphans for afternoon tea. One day, he gave each child 4 biscuits and had 2 biscuits left.

To give each child 6 biscuits, he would need another 22 biscuits. How many children were in his orphanage?

A: It seems like a difficult question at first glance. But if you understand the question, you can solve the question with simple steps. For easy understanding, draw a diagram to show the relationship of the number of children and the number of biscuits. Once you get the diagram correct, that means you have understood the question and it is easier for you to find the answer.

To give the children 6 biscuits each, he would need another 22 biscuits. That means 2 extra biscuits + 22 biscuits = 24 biscuits to be equally divided to the children.

2 + 22 = 24

There is an increase of 2 biscuits for each child, from 4 to 6.

6 – 4 = 2

To find the number of children, we divide the biscuits equally,

24 ÷ 2 = 12

There are 12 children in the orphanage.

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Checking the answer

When there are only 4 biscuits to each child,

(4 x 12) + 2 = 50

The owner has 50 biscuits initially, as there are 2 biscuits left.

He adds in another 22 biscuits and now he has 72 biscuits.

50 + 22 = 72

72 biscuits are divided equally by 12 children, each child gets 6 biscuits.

72 ÷ 12 = 6

Thus, the answer, 12 children, is correct.

Note to parents: If your child encounters difficult questions, asks him / her to check the answer. Checking the answer is a good way to reassure your child that he / she has answered the question correctly.

## Let’s Learn Mathematics (Primary Level)

Q: The sum of three numbers is 125. The smallest number is 25. The difference between the largest number and the smallest number is 35. Find the second largest number.

A: Method 1 — This is a common question for Primary students in Singapore. First you need to draw the diagram to show the relationship of the three numbers, based on the question. Once you get the diagram correct, that means you have understood the question and it is easier for you to find the answer.

To find the second largest number, you need to subtract the smallest number and the largest number from the sum.

The largest number = 25 + 35 = 60

The second largest number = 125 – 25 – 60

= 40

Method 2 — Use variables to represent the three numbers, a = the smallest number, b = the second largest number and c = the largest number.

a + b + c = 125

a = 25

c – a = 35

To find c, c = 35 + a = 35 + 25 = 60

To find b, b = 125 – a – c

= 125 – 25 – 60

= 40

Pros and cons of the above two methods

Method 1: Pro — This method is taught in school. As long as you get the diagram correct, you are not far from the correct answer.

Con — If the student cannot understand the question and unable to draw diagram, the student will leave the answer blank, which leads to zero mark. Many parents have not been taught about this method when they were studying. This is when both parents and students are at a loss because parents are unable to teach the children how to solve the question in “school way”.

Method 2: Pro — If you are good at algebra, this is a good method to use. Parents are able to teach the children how to solve the question. Often, the parents are afraid to teach the children this way because it is not the way how teachers teach in school. No worry, the students are actually encouraged to use different ways to solve problems, as long as they can find the correct answer.

Con — Algebra is a more complicated concept than diagram, thus some students may not get used to the idea of variables. If the students learn two methods at the same time, they may get confused.

Note to parents: Algebra will be taught in Secondary schools. Thus, if your children are able to solve the question using the model method, you may want to introduce the algebra method to your children as an alternative way to solve the question or to check whether the answer is correct.