Marvin bought a box of fruits. 30% of the fruits are apples and the rest are oranges. He realized that half of the apples were rotten and threw them away. He then bought some oranges and the number of oranges increased by 40%. After that, he found that there were 52 more fruits in the box. How many fruits were there in the box at first?
Solution
At first
Apples ----- 30%
Oranges ----- 70%
Threw half apples away, bought 40% more oranges,
Apples ----- 1/2 x 30% = 15%
Oranges ----- 70% + (40/100 x 70%) = 98%
Total ----- 15% + 98% = 113%
113% - 100% = 13% (increase in number of fruits)
13% ----- 52 fruits
1 % ----- 52% divided by 13 = 4
Number of fruits at first,
100% ----- 4 x 100 = 400
Answer: 400 fruits
Sunday, September 06, 2009
Anglo Chinese School 2007 PSLE Math Prelim Q44
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Question 45 : James is thinking of the smallest six - digits number. The first four digits are different. The first digit of this number starts with 6 and this number can also be divisible by 3, 4 and 5. Find this number. [5 marks]
Question 12 : James is thinking of the smallest six - digits number. The first four digits are different. The first digit of this number starts with 6 and this number can also be divisible by 3, 4 and 5. Find this number. [5 marks]
Solution:
List out the possible digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
The number : 6 _ _ _ _ _
This number is divisble by 3, 4 and 5.
Start with 5 first.
To be divisible by 5, we can derive the last digit to be "0"
Thus, the number is 6 _ _ _ _ 0
Next we can pick 0, 1 and 2 as they are the smallest.
6 0 1 2 _ 0
To be divisble by 4,
the digits in the tens place can be:
0, 2, 4, 6 and 8
To be divisble by 3, sum of digits must be multiple of 3.
6 + 0 + 1 + 2 + A + 0 = A + 9
Posibble values of A are:
0 and 6. 2, 4 and 8 are not possible as they are not divisible by 3.
Thus, these are the possible numbers:
601200, 601260
But, the question stated that the number have to be small.
Thus, the smallest number
is 601200.
Check back:
LCM = 3 x 4 x 5 = 60.
601200/60 = 10020 (correct)
A number has 6 digits. This number is divisible by 9 and the number starts with 8. The digits are all different. Find the smallest of this number.
Number: 8 _ _ _ _ _ _. Pick 0, 1, 2, 3. Then the number will become 8 0 1 2 3 _. To find the last digit, sum of digits must be multiple of 9. 8 + 0 + 1 + 2 + 3 + A = 14 + A. 14 + A = 18. Thus, A = 4. So, the smallest number will be 801234.
[b]Notes on Simple Divisibility - Part 1:[/b]
1)To be divisible by 2, the number must ends in 0, 2, 4, 6 or 8.
2)To be divisible by 5, the number must ends in 0 or 5.
3)To be divisible by 25, the number must ends in 25, 50 or 75.
4)To be divisible by 4 and by 25 (5 x 5) or (2 fives), the number must ends in 00.
5)To be divisible by 8 and by 125 (5 x 5 x 5) or (3 fives), the number must ends in 000.
6)To be divisible by 16 and by 625(5 x 5 x 5 x 5) or (4 fives), the number must end in 0000.
A number has 6 digits. This number is divisible by 9 and the number starts with 8. The digits are all different. Find the [b]largest[/b] of this number.
Solution:
List down all the digits:
0, 1, 2, 3, 4, 5, 6, 7, 8 and 9
The number: 8 _ _ _ _ _
since all digits are different and the number has to be large,
we will pick 9 for the second digit and 7 for the third digit, 6 and 5 for the fourth and fifth digits respectively.
Thus, we will have,
8 9 7 6 5 _.
To be divisible by 9, sum of the digits of the number is multiple of 9.
8 + 9 + 7 + 6 + 5 + A = 35 + A
Multiple of 9 which is higher than 35 is 36 (9 x 4).
35 + A = 36
A = 1
Thus, the number is 897651.
Note: We cannot choose 45 (9 x 5) is because the value of A will have 2 digits. Value of A can only be in 1 digit.
Additional notes with reference to Muffins' statement that angles in a 4-sided figure ALWAYS add up to 360 degree. A 4-sided figure is called a quadrilateral aka 4-sided polygon. A triangle is also known as a 3-sided polygon. As in P5/P6, you will learn that all sum of its angles of a triangle added up to 180 degree. For a quadrilateral, all sum of its angles is 360 degree. The rule is that all the sum of angles in a polygon with n sides, where n is 3 or more, is 180 degree x (n-2). Applying this rule to the other polygons, you will have a 5-sided polygon aka pentagon with its sum of angles equal to 540 degree. For hexagon(6 sides), sum = 720 degree. Trying working out for the rest like Heptagon (7-side), Octagon (8-side), Nonagon (9-side) and Decagon (10-side). BTW, a polygon is a closed figure made by joining line segments, where each line segment intersects exactly two others.
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