Common ratio questions: The question gives the
initial ratio of two items, and then certain quantities are added or subtract
from one of the quantity, resulting in a new set of ratio.
There are three methods given to solve for example 1.
The methods are (1) PROPORTION, (2) Models and (3) Comparison.
PROPORTION METHOD
Step1: Underline/Circle Keywords, and write value given
Step2: Put in the Ratio Value (u) and New Ratio Value
Step3: Equate New VALUE with NEW RATIO and solve (by cross multiply)
Example1
The ratio of the number of apples to the number of oranges in a basket was 5:4. After I gave away 21 apples, the ratio became 2:3. How many apples and oranges are there altogether in the basket in the end?
PROPORTION METHOD
Step1: Underline/Circle Keywords, and write
Step1: Underline/Circle Keywords, and write
A O
5 : 4
-21
New Ratio 2 : 3
Step2: Put Ratio Value (u) and New Ratio value
A O
5u : 4u
-21
New ratio 2 : 3
A's new value = 5u -21
O's new value = 4u (no change)
O's new value = 4u (no change)
Step3: Equate New VALUE with NEW RATIO and solve (by cross multiply)
A O
5u : 4u
-21
New ratio 2 : 3
New ratio value 5u - 21 : 4u
=> AA
OOO
=> AA AA AA
3A = 2O ( or cross multiply for 3A and 2O to equate value)
[Proportionally, A = 2/3O ]
=> AA
OOO
=> AA AA AA
OOO OOO
[Proportionally, A = 2/3O ]
3(5u – 21) = 2 x (4u)
15u – 63 = 8u
15u – 63 + 63 – 8u = 8u – 8u + 63
7u = 63
15u – 63 + 63 – 8u = 8u – 8u + 63
7u = 63
u = 9
OR
3A = 2O
(5u - 21) + (5u - 21) + (5u - 21) = 4u + 4u
15u – 63 = 8u
8u + 7u - 63 + 63 = 8u + 6
7u = 63
u = 9
OR
3A = 2O
(5u - 21) + (5u - 21) + (5u - 21) = 4u + 4u
15u – 63 = 8u
8u + 7u - 63 + 63 = 8u + 6
7u = 63
u = 9
In the end, there are 9u – 21 = 9 x 9 - 21 = 60 fruits
{{Using Model to explain A = 2/3 O [3A = 2O]
2/5 of A = 3/5 of O => Equal numerator
Find the common numerator(factor),
Common factor for 2 and 3 = 6
A = 2/5 x 3/3 = 6/10, B = 3/5 x 2/2 = 6/10
A = 6/15, B = 6/10
A = 5u – 21, B = 4u
3A = 2O
Solve for u as above.
}}
}}
Model Method
Step1: Keywords and Draw Model
A : O
5 : 4
Step2: Link the before and after ratio
Since there is no change to the number of orange
A : O
5 : 4
-21
2 : 3
Step3: Find common u by ‘equaling’
(a) Common Factor of 4 and 3 = 12, Divide both before and after into 12 parts
7u = 21
u = 3.
There are 20 x 3 = 60 fruits
There are 20 x 3 = 60 fruits
Comparison Method
Step1: Keywords and Identify Type of Compare
- Constant Unchanged / No change to one Constant : Orange
A : O
5 : 4
-21
2 : 3
Step2: ‘Equal’ the ‘Before’ and ‘After’ ratio number for orange
A : O
5x3 : 4 x3
-21
2x4 : 3 x4
Step3: Find u with Changed unit
A : O
15u : 12u
-21
8u : 12u
7u = 21
u = 3
Example2
PROPORTION METHOD
PROPORTION METHOD
The ratio of Annabel’s ribbons to Clara’s ribbons is 1:3. After Clara gave Annabel 9 ribbons, the ratio became 2:3. Find the number of ribbons Annabel had at first.
Step1: Underline/Circle Keywords, and write value given
A C
1 : 3
+9 -9
New ratio 2 : 3
Step2: Put Ratio Value (u) and New Ratio value
A C
1u : 3u
+9 -9
New ratio 2 : 3
A's new value = u + 9
C's new value = 3u - 9
C's new value = 3u - 9
Step3: Equate New VALUE with NEW RATIO and solve (by cross multiply)
A C
1u : 3u
+9 -9
+9 -9
New ratio 2 : 3
New ratio value u+9 : 3u-9
=> AA
CCC
=> AA AA AA
CCC CCC
3(u + 9) = 2(3u - 9) or
3u – 3u +27 + 18 = 6u – 3u - 18 + 18 3u + 27 = 6u - 18
45 = 3u 3u + 27 + 18 = 6u – 18 + 18
3u = 45 3u + 45 = 6u
u = 15 3u + 45 = 3u + 3u
3u = 45
Annabel had 15 ribbons at first.
Model Method
Step1: Keywords and Draw Model
Step2: Link the before and after ratio
- There is no change to the total value
Step3: Find common u by ‘equaling’
(a) Common Factor of 4 and 3 = 12, Divide both before and after into 12 parts
3u = 9, u = 3
Annabel had 5 x 3 = 15 ribbons at first
Example3
At first, the ratio of Alan’s marbles to Kevin’s marbles was 3:4. After Alan bought another 9 marbles and Kevin lost 18 marbles, the ratio became 3:2. Find the number of marbles Alan had at first.
Step1: Underline/Circle Keywords, and write value given
A K
3 : 4
+9 -18
New ratio 3 : 2
Step2: Put Ratio Value (u) and New Ratio value
A K
3u : 4u
+9 -18
3 : 2
A's new value = 3u + 9
A's new value = 3u + 9
C's new value = 4u - 18
A K
3u : 4u
+9 -18
New ratio 3 : 2
New ratio 3 : 2
New ratio value 3u+9 : 4u-18
=> AAA
KK
=> AAA AAA
KK
=> AAA AAA
KK KK KK
2A = 3K ( or cross multiply for 2A and 3K to equate value)
2(3u + 9) = 3(4u - 18) or
6u – 6u +18 + 54 = 12u – 6u – 54 + 54 6u + 18 + 54 = 12u – 54 + 54
72 = 6u 6u + 72 = 12u
6u = 72 6u + 72 = 6u + 6u
u = 12 6u = 72, u = 12
Alan had 3 x 12 = 36 marbles at first.
Model Model
Step1: Keywords and Draw Model
Step2: Link the before and after ratio
- Both change in quantities
Step3:
Relate quantity to U
u + 3 = 2u – 9
u = 3 + 9
= 12
Alan had 3 x 12 = 36 marbles at first.
Example 4
The ratio of the number of pears to the number of mango was 3:2. After 10 pears were taken away, the ratio became 2:3. How many fruits are there altogether in the end?
PROPORTION METHOD
PROPORTION METHOD
Step1: Underline/Circle Keywords, and write value given
P M
3 : 2
-10
New 2 : 3
Step2: Put Ratio Value (u) and New Ratio value
P M
3u : 2u
-10
New ratio 2 : 3
P's new value = 3u - 10
M's new value = 2u (no change)
Step3: Equate New VALUE with NEW RATIO and solve (by cross multiply)M's new value = 2u (no change)
P M
3u : 2u
-10
New ratio 2 : 3
New Value 3u - 10 : 2u
=> PP
MMM
=> PP PP PP
MMM MMM
3P = 2M
New Value 3u - 10 : 2u
=> PP
MMM
=> PP PP PP
MMM MMM
3(3u – 10 ) = 2 x (2u)
9u – 30 = 4u
9u - 4u -30 + 30 = 4u - 4u + 30
5u = 30
u = 6
In the end, there are 5u – 10 = 30 -10 = 20 fruits
Example5
The ratio of the number of woman to the number of men in a concert is 2 : 3. 65 women left and the ratio of the number of women to the men became 1:4. Find the total number attended the concert.
PROPORTION METHOD
PROPORTION METHOD
Step1: Underline/Circle Keywords, and write value given
W M
2 : 3
-65
1 : 4
Step2: Put Ratio Value (u) and New Ratio value
W M
2u : 3u
-65
New ratio 1 : 4
W's new value = 2u - 65
M's new value = 3u (no change)
M's new value = 3u (no change)
Step3: Equate New VALUE with NEW RATIO and solve (by cross multiply)
W M
2u : 3u
-65
New ratio 1 : 4
New ratio 1 : 4
New ratio value 2u -65 : 3u
=> W
MMMM
=> W W W W
M M M M
4(2u - 65) = 3u or
8u - 260 = 3u 8u – 260 + 260 = 3u + 260
8u – 3u -260 + 260 = 3u – 3u + 260 3u + 5u = 3u + 260
5u = 260 5u = 260
u = 52 u = 52
Total number = 5 x 52 = 260
Example6
The ratio of the number of red balls to the blue balls was 3 : 7. After equal number of red and blue balls were given away. In the end, the ratio of the red balls left to the blue balls was 5:13. There were 20 red balls in the end. How many red balls were given away?
Step1: Underline/Circle Keywords, and write information
R B
3 : 2
same number taken
5 : 13
20 red pens for 5part, 1 part = 20/5 = 4
Black pen = 4 x 13 = 52
Step2: Put in the Ratio Value (u)
R B
3u : 7u
same number taken
20 : 52
Step3: Since equal number of balls are taken away
3u – 20 = 7u – 52 3u – 20 + 52 = 7u – 52 + 52
7u – 3u = 52 – 20 3u +32 = 3u + 4u
4u = 32 4u = 32
u = 8
3x 8 – 20 = 4 red balls are given away.
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