Primary School Leaving Examination, Mathematics Question
The Primary School Leaving Examination (PSLE) is the national examination in Singapore taken by Primary 6 pupils. Over the years, the Mathematics paper is well known to contain very challenging questions for students for that level.
Here is one question and its possible solution from a recent PSLE Mathematics paper, which I believe is from the 2019 edition:
Of course, the given solution for part (c) is perfectly valid. However, I tend to dislike having to extend the table and seeing so many numbers. Therefore, I propose another approach for solving part (c). I shall mention upfront that my approach is not necessarily easier than the solution given above, but it is still another way of solving this problem.
First, note that Fig 250 has 250 rows of triangles of alternating colour, and the last row of Fig 250 is grey since 250 is an even number. Thus, we join one copy Fig 250 with an upside-down copy of Fig 250, as shown in the following diagram.
In this diagram, it is easy to see that there are 250 ÷ 2 = 125 rows of grey triangles, and each row has the same number of grey triangles. We now consider the last row of grey triangles in this diagram. The number of triangles in this last row is (number of triangles in last row of Fig 250) + (number of triangles in second row of Fig 250). The number of triangles in second row of Fig 250 is 3. So we need to find the number of triangles in the last row of Fig 250.
To do this, we consider the number of triangles in the first row, second row, third row, and so on. The sequence of numbers is 1, 3, 5, 7, …. As the difference between consecutive numbers in this sequence is 2, the 250th number in this sequence can be calculated as 1 + (2 × 249) = 499. Therefore, the last row in Fig 250 has 499 triangles.
Hence, the number of triangles in each grey row in the diagram is 499 + 3 = 502. The total number of grey triangles in the diagram is 125 × 502 = 62750. Given that the diagram is two copies of Fig 250, this follows that the number of grey triangles in Fig 250 is 62750 ÷ 2 = 31375.
The answer for part (c), which is the percentage of grey triangles in Fig 250, is 31375 ÷ 62500 = 50.2%.
Here is one question and its possible solution from a recent PSLE Mathematics paper, which I believe is from the 2019 edition:
Of course, the given solution for part (c) is perfectly valid. However, I tend to dislike having to extend the table and seeing so many numbers. Therefore, I propose another approach for solving part (c). I shall mention upfront that my approach is not necessarily easier than the solution given above, but it is still another way of solving this problem.
First, note that Fig 250 has 250 rows of triangles of alternating colour, and the last row of Fig 250 is grey since 250 is an even number. Thus, we join one copy Fig 250 with an upside-down copy of Fig 250, as shown in the following diagram.
In this diagram, it is easy to see that there are 250 ÷ 2 = 125 rows of grey triangles, and each row has the same number of grey triangles. We now consider the last row of grey triangles in this diagram. The number of triangles in this last row is (number of triangles in last row of Fig 250) + (number of triangles in second row of Fig 250). The number of triangles in second row of Fig 250 is 3. So we need to find the number of triangles in the last row of Fig 250.
To do this, we consider the number of triangles in the first row, second row, third row, and so on. The sequence of numbers is 1, 3, 5, 7, …. As the difference between consecutive numbers in this sequence is 2, the 250th number in this sequence can be calculated as 1 + (2 × 249) = 499. Therefore, the last row in Fig 250 has 499 triangles.
Hence, the number of triangles in each grey row in the diagram is 499 + 3 = 502. The total number of grey triangles in the diagram is 125 × 502 = 62750. Given that the diagram is two copies of Fig 250, this follows that the number of grey triangles in Fig 250 is 62750 ÷ 2 = 31375.
The answer for part (c), which is the percentage of grey triangles in Fig 250, is 31375 ÷ 62500 = 50.2%.