Given that a : b = 3 : 5 and b : c = 4 : 7, find the ratio a : b : c in its simplest form.
Worked Solution & Answer
Match b = 20: a : b : c = 12 : 20 : 35. B1
Answer: 12 : 20 : 35
Combining ratios — Notes & Worked Examples Show / Hide
Explanation:
To combine two separate ratios such as a : b and b : c into a single three-part ratio a : b : c, identify the item that appears in both ratios (here, b). Find the lowest common multiple (LCM) of the two numbers representing that shared item.
Multiply all parts of each ratio so that the shared item has the exact same value in both ratios, then write them together as a : b : c. Once combined, you can add the parts to solve total-amount problems or compare any two quantities directly.
Example 1: Given that a : b = 2 : 3 and b : c = 4 : 5, find the ratio a : b : c in its simplest form.
Solution:
Match b using the LCM of 3 and 4, which is 12:
• Multiply a : b = 2 : 3 by 4 → 8 : 12
• Multiply b : c = 4 : 5 by 3 → 12 : 15
Combine the three parts together:
Answer: a : b : c = 8 : 12 : 15
Example 2: In a bag of 66 counters, red : blue = 3 : 2 and blue : green = 5 : 4. How many blue counters are there?
Solution:
Match blue (LCM of 2 and 5 is 10):
• R : B = 15 : 10 and B : G = 10 : 8 → 15 : 10 : 8
Total parts = 15 + 10 + 8 = 33 parts
1 part = 66 ÷ 33 = 2 counters; Blue = 10 × 2 = 20
Answer: 20 blue counters
Cats and dogs in a shelter are in the ratio 3:8
There are 30 more dogs than cats
How many animals are there in total?
There are 30 more dogs than cats
How many animals are there in total?
Worked Solution & Answer
Add the ratio parts for the total parts (or divide by the known part to find the value of 1 part), then multiply.
66 animals
66 animals
Answer: 66 animals
In a village
[INDENT]the number of houses and the number of flats are in the ratio 7 : 4
[INDENT]the number of flats and the number of bungalows are in the ratio 8 : 5
There are 50 bungalows in the village.
How many houses are there in the village?
[INDENT]the number of houses and the number of flats are in the ratio 7 : 4
[INDENT]the number of flats and the number of bungalows are in the ratio 8 : 5
There are 50 bungalows in the village.
How many houses are there in the village?
Worked Solution & Answer
Houses : Flats : Bungalows = 14 : 8 : 5. 50 ÷ 5 = 10 M1
Flats = 8 × 10 = 80 M1
Houses = 14 × 10 = 140. A1
Flats = 8 × 10 = 80 M1
Houses = 14 × 10 = 140. A1
Answer: 140
Cormac has some sweets in a bag.
The sweets are lime flavoured or strawberry flavoured or orange flavoured.
In the bag
[INDENT]number of lime flavoured sweets : number of strawberry flavoured sweets : number of orange flavoured sweets = 9 : 4 : x
Cormac is going to take at random a sweet from the bag.
The probability that he takes a lime flavoured sweet is 37
Work out the value of x.
The sweets are lime flavoured or strawberry flavoured or orange flavoured.
In the bag
[INDENT]number of lime flavoured sweets : number of strawberry flavoured sweets : number of orange flavoured sweets = 9 : 4 : x
Cormac is going to take at random a sweet from the bag.
The probability that he takes a lime flavoured sweet is 37
Work out the value of x.
x =
Worked Solution & Answer
9 / (9 + 4 + x) = 3 / 7 = 9 / 21 M1
⇒ 9 + 4 + x = 21 M1
⇒ x = 21 – 13 = 8 A1
⇒ 9 + 4 + x = 21 M1
⇒ x = 21 – 13 = 8 A1
Answer: x = 8
In another bag, Bag C, the ratio of red counters to yellow counters is 3 : 4.
If 3 of the red counters are removed from Bag C, the ratio of red counters to yellow counters is 3 : 5.
How many **yellow** counters are in Bag C?
If 3 of the red counters are removed from Bag C, the ratio of red counters to yellow counters is 3 : 5.
How many **yellow** counters are in Bag C?
Worked Solution & Answer
Let the initial numbers of red and yellow counters be 3x and 4x M1
When 3 red counters are removed:
(3x − 3) / (4x) = 3 / 5 M1
5(3x − 3) = 12x
15x − 15 = 12x ⇒ 3x = 15 ⇒ x = 5.
Number of yellow counters = 4x = 4 × 5 = 20. A1
When 3 red counters are removed:
(3x − 3) / (4x) = 3 / 5 M1
5(3x − 3) = 12x
15x − 15 = 12x ⇒ 3x = 15 ⇒ x = 5.
Number of yellow counters = 4x = 4 × 5 = 20. A1
Answer: 20
Here is a quadrilateral.
Show that b = x
a = 90° and a : b = 5 : 3
x : y = 1 : 3
x : y = 1 : 3
Worked Solution & Answer
From a : b = 5 : 3 and a = 90°: M1
b = 35 × 90° = 54°
Sum of angles in a quadrilateral = 360°:
a + b + x + y = 360° M1
90° + 54° + x + y = 360° ⇒ x + y = 216°
From x : y = 1 : 3, total parts = 4:
x = 14 × 216° = 54°
Since b = 54° and x = 54°, b = x. A1
b = 35 × 90° = 54°
Sum of angles in a quadrilateral = 360°:
a + b + x + y = 360° M1
90° + 54° + x + y = 360° ⇒ x + y = 216°
From x : y = 1 : 3, total parts = 4:
x = 14 × 216° = 54°
Since b = 54° and x = 54°, b = x. A1
Answer: Answer Proof that b = x = 54°
A cylinder, C, and a sphere, S, each have radius r
C has height h
C has height h
(a) @@TABULAR0@@
volume of C = volume of S
Work out the ratio r : h
You **must** show your working. [3 marks]
volume of C = volume of S
Work out the ratio r : h
You **must** show your working. [3 marks]
Answer
Worked Solution & Answer
(a)
πr²h = 4/3 πr³ M1
⇒ h = 4/3 r M1
⇒ 3h = 4r
⇒ r : h = 3 : 4. A1
⇒ h = 4/3 r M1
⇒ 3h = 4r
⇒ r : h = 3 : 4. A1
Answer (a): Answer 3 : 4
Helen has 80 books to sell.
Each book is Fiction or Non-fiction.
The ratio of the number of Fiction books to the number of Non-fiction books is 3:1
Each book has a normal price of £10
Helen reduces the price of all the Non-fiction books.
Each book is Fiction or Non-fiction.
The ratio of the number of Fiction books to the number of Non-fiction books is 3:1
Each book has a normal price of £10
Helen reduces the price of all the Non-fiction books.
Helen sells all 80 books.
Work out the total amount of money Helen will receive.
Work out the total amount of money Helen will receive.
£
Worked Solution & Answer
3 + 1 = 4 parts M1
Non-fiction = 80 ÷ 4 = 20 books, Fiction = 60 books M1
Fiction = 60 × £10 = £600 M1
Non-fiction = 20 × £5 = £100
Total = £600 + £100 = £700 A1
Non-fiction = 80 ÷ 4 = 20 books, Fiction = 60 books M1
Fiction = 60 × £10 = £600 M1
Non-fiction = 20 × £5 = £100
Total = £600 + £100 = £700 A1
Answer: £ 700
Tom and Adam have a total of 240 stamps.
The ratio of the number of Tom's stamps to the number of Adam's stamps is 3 : 7
Tom buys some stamps from Adam.
The ratio of the number of Tom's stamps to the number of Adam's stamps is now 3 : 5
How many stamps does Tom buy from Adam?
You must show all your working.
The ratio of the number of Tom's stamps to the number of Adam's stamps is 3 : 7
Tom buys some stamps from Adam.
The ratio of the number of Tom's stamps to the number of Adam's stamps is now 3 : 5
How many stamps does Tom buy from Adam?
You must show all your working.
Worked Solution & Answer
Before: 240 ÷ 10 = 24, so Tom has 3 × 24 = 72 stamps (Adam has 168) M2
After: 240 ÷ 8 = 30, so Tom has 3 × 30 = 90 stamps (Adam has 150) M1
Stamps bought = 90 – 72 = 18 A1
After: 240 ÷ 8 = 30, so Tom has 3 × 30 = 90 stamps (Adam has 150) M1
Stamps bought = 90 – 72 = 18 A1
Answer: 18
Rizwan writes down three numbers a, b and c
a : b = 1 : 3
b : c = 6 : 5
a : b = 1 : 3
b : c = 6 : 5
(a) Find a : b : c [2 marks]
(b) Express a as a fraction of the total of the three numbers a, b and c [2 marks]
(b) Find m : p [2 marks]
Worked Solution & Answer
(a)
a : b = 2 : 6 M1
⇒ a : b : c = 2 : 6 : 5 A1
⇒ a : b : c = 2 : 6 : 5 A1
Answer (a): 2 : 6 : 5
(b)
2 / (2 + 6 + 5) = 2/13 M1 A1
Answer (b): 213
(b)
p = 5(2m) = 10m M1
⇒ m : p = 1 : 10 A1
⇒ m : p = 1 : 10 A1
Answer (b): 1 : 10