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Ratio (Maths 1 Shinglee pg243) Example

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 8x12 =96

Mock T1/P2 Q7

There are 315 pupils in a school.
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) = 8x21 = 168

Ratio question

Stumped?

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

Ratio Question Arthur Yusuf and Bill

Arthur, Yusuf and Bill had some stamps in the ratio 3:5:6. When Yusuf gave some stamps to Arthur and Bill gave 42 more than what Yusuf gave to Arthur, they all had the same number of stamps each. How many more stamps than Arthur did Yusuf have at first?

Before: A:Y:B = 3:5:6 (total 14 units)
After: A:Y:B = 1:1:1 (total 3 units)

Since the total number of stamps remain unchanged, use equivalent ratios for the total units. (LCM of 14 and 3 = 42)

so Before A:Y:B = 9:15:18 (total 42 units)
After A:Y:B =14:14:14 (total 42 units)

We see that A has gained 5 units, Y lost 1 unit and B lost 4 units

B lost 3 more units than Y (4-1).
From the question Bill gave 42 more stamps than Yusuf, so 1 unit = 14 stamps (42/3)

At the beginning, Arthur had 9 units and Yusuf 15 units (6 more units)
So Yusuf had 6x14=84 stamps more than Arthur

CM5 Unit4 Q3

Useful techniques for Word Problems

using the units method:

Question: (When One item remains the same)
1. The ratio of the weight of the mutton to the beef is 3:4. When 57kg of the beef was sold, the ratio of the weight of the mutton to the beef becomes 7:3
(a) Find the weight of the beef before it was sold.
(b) Find the weight of the mutton.

Solution:

Before: M:B = 3:4
After: M:B = 7:3

Since the mutton remains the same, the "3" before should be the same quantity as the "7" after,
so, expressing it in equivalent ratios,

Before: M:B = 3:4 or 21:28
After: M:B = 7:3 or 21:9

Comparing before and after, B has been reduced by 19 units (28-9)
19 units = 57kg
so 1 unit = 3 kg

(a) Beef before it was sold was 28 units or 28x3 ==> 84kg
(b) Mutton (no change) weighed 21x3 ==> 63kg


Question: (When the total remains the same)
The ratio of Jessie's money to Katie's money is 3:2. If Jessie gives Katie $3.50, the ratio of their money will become 1:3
(a) How much money does Jessie have (after giving)?
(b) How much money does Katie have (after receiving)?

Before: J:K = 3:2 (total 5 units)
After: J:K = 1:3 (total 4 units)

Since J gave K money, the total money before and after remains the same, so we put it in equivalent terms. In this case the LCM of 5 and 4 gives 20.

Before: J:K = 3:2 or 12:8 (total 20 units)
After: J:K = 1:3 or 5:15 (total 20 units)

We can see that J has decreased by 7 units and K increased by 7 units
7 units = $3.50
1 unit = $0.50

(a) Jessie's money (after giving) is 5 units or $2.50
(b) Katie's money (after receiving) is 15 units or $7.50

Question: (When the difference remains the same)
The ratio of John's age to his friend's age is 2:5. In 7 years the ratio of their ages will be 3:4. What is John's age now?

Now: J:F = 2:5 (difference 3 units)
Later J:F = 3:4 (difference 1 unit)

Since the age difference is the same, we want to get the same equivalent units for the difference:

Now: J:F = 2:5 (difference 3 units)
Later J:F = 3:4 or 9:12 (difference 3 units)
We can see John and friend have increased by 7 units.
7 units = 7 years
1 unit = 1 year

John's age is now 2.

Good reference:

Earlier than Pascal's Triangle

From http://milan.milanovic.org/math/english/fibo/fibo0.html

The so called 'Pascal' triangle was known in China as early as 1261. In '1261 the triangle appears to a depth of six in Yang Hui and to a depth of eight in Zhu Shijiei in 1303. Yang Hui attributes the triangle to Jia Xian, who lived in the eleventh century.