number pattern 6,9,14,21…
Posted by: pokopants on: May 21, 2009
The numbers above increase in increments as follow: 3,5,7,9…
We see that squares also increase in this pattern:
1,4,9,16,25,36,49… ==> increase by 3,5,7,9,…
So pattern is (n-squared + 5)
since it is actually
(1+5), (4+5),(9+5),(16+5),(25+5) …
simplified simultaneous equations
Posted by: pokopants on: May 17, 2009
- In: ratio
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Question: There were 120 red and blue marbles in a bag at first. 25% of the red marbles and 2/3 of the blue marbles were given away. In the end there were 60 red and blue marbles. How many red marbles were there at first?
To simplify this question for my P4 daughter who has yet to learn about percentages or ratio:
25% = 1/4. At the end red marbles = 3/4 (end 3 units, start 4 units)
| Red marbles at first: | ♣ | ♣ | ♣ | ♣ |
| Red marbles at end | ♣ | ♣ | ♣ |
Blue marbles at end = 1/3 left (end 1 units, start 3 units)
| Blue marbles at first: | ♥ | ♥ | ♥ |
| Blue marbles at end: | ♥ |
We know that at first the total was 120, so
♣♣♣♣+♥♥♥=120
At the end the total is 60, or
♣♣♣+♥=60
Make the blue marbles the same units (3 sets), so
♣♣♣♣♣♣♣♣♣+♥♥♥=180
Now if we compare the 2, we see that
♣♣♣♣♣ = 180-120 = 60
so ♣ = 12 and ♥= 60-3×12= 24
At beginning, red marbles = ♣♣♣♣ or 4×12=48
This question is also easily solved using the units method by comparing the before and after ratios.
(CMaths Ammiel Wan Unit 10 modified – great book!)
candle ratio question
Posted by: pokopants on: May 17, 2009
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Legend has it that this question stumped all but 5 students in a school when it first appeared in an exam paper a few years back.
Question:
Two candles, one of which is one cm longer than the other, are lit. The longer one is lit at 4:30pm and the shorter one at 6:00 pm. At 8:30 they are both the same length. The longer one burns out at 10:30pm and the shorter one at 10:00pm. How long was each candle originally?
Solution:
Time taken for L (longer candle) = 6 hrs (4:30 to 10:30pm)
Time for S (shorter candle) = 4 hrs (6 to 10pm)
At 8:30 they are at the same length.
Time taken for L to burn out –> 2 hours ===> Candle is 2/6 or 1/3 long
Time taken for S to burn out –> 1.5 hours ===> Candle is 1.5/4 or 3/8 long
Since the 2 lengths are the same at 8:30, this means that 1/3 of L is the same as 3/8 of S
or 3/9 of L is the same as 3/8 of S
so ratio of L to S is 9:8
L is longer by one unit. We are told L is longer by 1 cm, so L is 9cm and S 8 cm long.
penalty marks
Posted by: pokopants on: May 16, 2009
Question:
Sally answered 100 questions in a test. She gets 1 mark for every correct answer. 2 marks are deducted for every wrong answer. She scored 79 marks. How many questions did she answer wrongly?
Method 1:
Method 2:
Correct answer –> 1 marks
Wrong answer –> deduct 2 marks ==> total penalty 3 marks (2+1)
100-79=21
21/3 = 7
She answered 7 questions wrongly.
squares and their ending numbers
Posted by: pokopants on: May 16, 2009
We are familiar with this pattern: 1,4,9,16,25,36,49,64,81,100…
Start with square of 11 ==> 121,144,169,196,225,256,289,324,361,400…
Ending digits for 1st set : 1,4,6,9,6,5,6,9,4,1,0
Ending digits for 2nd set: 1,4,6,9,6,5,6,9,4,1,0
This pattern repeats itself.
So, numbers ending with 2,3,7,8 cannot be perfect squares!
Solving word problems involving 2 pairs of ratios
Posted by: pokopants on: May 16, 2009
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Just back from Maths workshop by Ernest Wong this afternoon.
Useful bags and boxes method (Raffles Maths Trick Vol 1 Trick 2)
for solving word problems involving 2 pairs of ratios.
http://www.moe.gov.sg/education/syllabuses/sciences/files/maths-primary-2007.pdf
Question:
The ratio of the number of apples and oranges in a shop is 10:3.
When 50 apples and 40 oranges were added, the new ratio became 3:1.
How many apples were in the shop at first?
Solution:
Put the apples and oranges in bags and boxes.
Each bag contains the SAME number of fruits (apples or oranges)
Each box containsthe SAME number of fruits (apples or oranges)
Each bag many contain more or less fruits than each box (Not Equal)
At first ratio of apples to oranges was 10:3 (Bags)
Finally ratio of apples to oranges was 3:1 (Boxes)
Let there be 10 bags of apples and 3 bags of oranges
Let there be 3 boxes of apples and 1 box of oranges finally.
Fruits Apple Oranges
Bags 10 3
Fruits added +50 +40
Boxes 3 1
Apples ===> 10 bags + 50 –> 3 boxes
Oranges ===> 3 bags + 40 –> 1 box
or 9 bags + 120–>3 boxes
Boxes have same amount of apples and oranges,
so 9 bags + 120 = 10 bags +50 ===> 1 bag = 70
Apples at first = 10 bags or 700.
Answer check:
At first : Apples = 700 Oranges = 210 (Ratio 10:3 ok)
Later Apples = 750 Oranges =250 (Ratio 3:1 ok)
Hello world!
Posted by: pokopants on: May 15, 2009
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Ratio (Maths 1 Shinglee pg243) Example
Posted by: pokopants on: May 12, 2009
- In: ratio | S1 example
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The ratio of the number of apples to oranges in Stall A and Stall B are 2:5 and 5:9 respectively. If the total number of fruits in Stall A is twice that in Stall B,
(a) find the ratio of the number of apples in Stall A to the number of apples in Stall B;
(b) find the number of oranges left in Stall A after 84 oranges are sold from Stall A and the ratio of apples to oranges becomes 3:4 in Stall A.
This example in the Sec 1 text was solved using Algebra. Here we try the units method:
(a)
Stall A Apples:Oranges = 2:5 (total 7 units) or 8:20 (28 units – fruits in Stall A double Stall B’s)
Stall B Apples:Oranges = 5:9 (total 14 units)
Ratio of Apples (Stall A) to Apples (Stall B) = 8:5
(b)
Before : Stall A A:O = 2:5 or 6:15 (A stays same, so make the same units before and after)
After: A:O = 3:4 or 6:8
Oranges -7 units or -84
7 units = 84, 1 unit = 12
Oranges (after) = 8 units or 8×12 =96
Mock T1/P2 Q7
Posted by: pokopants on: May 7, 2009
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3/7 of the boys and 3/4 of the girls do not wear watches.
The number of boys who wear watches is twice the number of girls who wear watches. How many girls are there altogether?
Model method:
Those who wear watches – 4/7 of the boys and 1/4 of the girls.
4/7of the boys = 1/4 of the girls
Using model, draw 2 units for boys and 1 for girls.
For girls, add 3 more units (total of 1/4+3/4)
For boys add 1.5 units (total of 4/7 and 3/7)
Altogether 15 units = 315
1 unit =21
Girls = 8 units = 168
Units method:
Boys==> Wear: No wear = 4:3 ==> 4:3
Girls==> Wear: No wear = 1:3 ==> 2:6
Since the boys who wear = twice the girls who wear, make the ratio equivalent (girls = 1/2 units of 4=2)
Total units = 4+2+3+6=15
15 units = 315
1 unit = 21
Girls total 8 units (2+6) = 8×21 = 168
Ratio question
Posted by: pokopants on: May 4, 2009
- In: ratio
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Question: Class A and Class B have the same number of pupils. The ratio of boys in Class A to the number of boys in Class B is 3:2. The ratio of the number of girls in Class A to the number of girls in Class B is 3:5. Find the ratio of the number of boys to the number of girls in Class A.
For those who know Algebra, this can be solved, albeit in a convulated way, by expressing Class A and Class B in terms of the other, i.e.
Class A = Boys(A) + Girls(A) or C(A) = B(A)+G(A)
Given:
B(A):B(B)=3:2 and G(A):G(B)=3:5
C(A) = 3/2 B(B) + 3/5 G(B)
C(B) = 2/3 B(A) + 5/3 G(A)
Since total in class A and B are equal,
equate the 2.
but there must be an easier method!!!
Try using units method:
Boys(A):Boys(B) = 3:2
Girls(A):Girls(B) = 3:5
Since Class A = Class B in size, Boys(A) + Girls (A) = Boys (B) + Girls (B)
Boys(A):Boys(B) = 3:2 or 6:4 (double)
Girls(A):Girls(B) = 3:5 ….. 3:5
Now units of Class A = units of Class B (9 units)
Now Boys (A) : Girls (A) = 6:3 or 2:1
