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16 Dec 2021, 08:28
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
58% (03:19) correct
42%
(03:22)
wrong
based on 178
sessions
History
Date
Time
Result
Not Attempted Yet
On Sunday, in Alexa’s stable, the ratio of the number of ponies to the number of horses was 4:5. On Monday, she bought 35 more ponies and horses, then the ratio of ponies to horses became 5:8. Assuming that no animal left the stable or died and stable had ponies and horses only, what is the least number of horses that were there in the stable on Sunday?
A.35
B.55
C.60
D.65
E.70
A.35
B.55
C.60
D.65
E.70
24 Dec 2021, 10:41
Kudos
Bookmarks
Here is the OE
Solution:
Step 1: Understand Question Statement
• On Sunday and Monday, only horses and ponies were there in Alexa’s stable.
• On Sunday, the ratio of the number of ponies to the number of horses was 4 :5.
• On Monday, 35 more ponies and horses came to the stable.
• The new ratio of the number of ponies to horses becomes 5:8
• No animals left the stable or died on Sunday and Monday.
We need to find the least number of horses that were in Alexa’s stable on Sunday.
Step 2: Define Methodology
• Let’s assume that the numbers of ponies and horses were 4x and 5x respectively, on Sunday.
• Assuming that the m horses came on Monday.
• The total number of horses on Monday = \(5x+m\)
• Total number of animals in stable on Monday = \(9x+35\)
• Since ratio of the number ponies to horses on Monday = \(\frac{5}{8}\)
• The ratio of the number of horses to total animals in stable =\( \frac{8}{8+5}=\frac{8}{13}=\frac{5x+m}{9x+35} \)
• We will solve the above equation to find the value of number of horses on Sunday.
Step 3: Calculate the final answer
• \(\frac{5x+m}{9x+35}=\frac{8}{13} \)
• \(65x+13m=72x+280 \)
⟹\(m=\frac{280+7x}{13} =\frac{280}{13}+\frac{7x}{13}=21+\frac{7}{13}+\frac{7x}{13}=21+\frac{7\left(x+1\right)}{13} ……. (i) \).
Since m is the number of horses, it must be a positive integer.
So, the least value of x for which m is an integer is = \((x+1) = 13⟹x = 12\)
Hence, \(x= 12\) and the least number of horses on Sunday = \(5\ast12=60\)
Thus, the correct answer is Option C.
Solution:
Step 1: Understand Question Statement
• On Sunday and Monday, only horses and ponies were there in Alexa’s stable.
• On Sunday, the ratio of the number of ponies to the number of horses was 4 :5.
• On Monday, 35 more ponies and horses came to the stable.
• The new ratio of the number of ponies to horses becomes 5:8
• No animals left the stable or died on Sunday and Monday.
We need to find the least number of horses that were in Alexa’s stable on Sunday.
Step 2: Define Methodology
• Let’s assume that the numbers of ponies and horses were 4x and 5x respectively, on Sunday.
• Assuming that the m horses came on Monday.
• The total number of horses on Monday = \(5x+m\)
• Total number of animals in stable on Monday = \(9x+35\)
• Since ratio of the number ponies to horses on Monday = \(\frac{5}{8}\)
• The ratio of the number of horses to total animals in stable =\( \frac{8}{8+5}=\frac{8}{13}=\frac{5x+m}{9x+35} \)
• We will solve the above equation to find the value of number of horses on Sunday.
Step 3: Calculate the final answer
• \(\frac{5x+m}{9x+35}=\frac{8}{13} \)
• \(65x+13m=72x+280 \)
⟹\(m=\frac{280+7x}{13} =\frac{280}{13}+\frac{7x}{13}=21+\frac{7}{13}+\frac{7x}{13}=21+\frac{7\left(x+1\right)}{13} ……. (i) \).
Since m is the number of horses, it must be a positive integer.
So, the least value of x for which m is an integer is = \((x+1) = 13⟹x = 12\)
Hence, \(x= 12\) and the least number of horses on Sunday = \(5\ast12=60\)
Thus, the correct answer is Option C.
29 Jan 2023, 11:45
Kudos
Bookmarks
Given: On Sunday, in Alexa’s stable, the ratio of the number of ponies to the number of horses was 4:5. On Monday, she bought 35 more ponies and horses, then the ratio of ponies to horses became 5:8.
Asked: Assuming that no animal left the stable or died and stable had ponies and horses only, what is the least number of horses that were there in the stable on Sunday?
Let the number of ponies and horses in Alexa's stable be p & h respectively.
On Sunday, in Alexa’s stable, the ratio of the number of ponies to the number of horses was 4:5.
p/h = 4/5
p = 4k
h = 5k
Total = 4k + 5k = 9k
On Monday, she bought 35 more ponies and horses, then the ratio of ponies to horses became 5:8.
Total = 9k + 35
Horses = 5k + h
Ponies = 4k + (35-h)
Ratio of horses/Total = 8/13 = (5k+h)/(9k+35)
72k + 280 = 65k + 13h
7k + 280 = 13h
k = (13h - 280)/7 = 13h/7 - 40
h = 4; k = 12
Minimum number of horses on Sunday = 5k = 60
IMO C
Asked: Assuming that no animal left the stable or died and stable had ponies and horses only, what is the least number of horses that were there in the stable on Sunday?
Let the number of ponies and horses in Alexa's stable be p & h respectively.
On Sunday, in Alexa’s stable, the ratio of the number of ponies to the number of horses was 4:5.
p/h = 4/5
p = 4k
h = 5k
Total = 4k + 5k = 9k
On Monday, she bought 35 more ponies and horses, then the ratio of ponies to horses became 5:8.
Total = 9k + 35
Horses = 5k + h
Ponies = 4k + (35-h)
Ratio of horses/Total = 8/13 = (5k+h)/(9k+35)
72k + 280 = 65k + 13h
7k + 280 = 13h
k = (13h - 280)/7 = 13h/7 - 40
h = 4; k = 12
Minimum number of horses on Sunday = 5k = 60
IMO C
