A shop sells two variants of chocolates - one that costs $3 and the other that costs $5. If the shop sold $108 chocolates on a given day, how many different combinations of (number of $3 sold, number of $5 sold) exist?
Correct Answer : Choice D. 8
Let the shop sell x numbers of the $3variant and y numbers of the $5 variant.
So, 3x + 5y = 108.
The only constraint to keep in mind is that both x and y are non-negative integers.
We can rewrite the equation as x = (108 - 5y)/3
So, (108 - 5y) should be divisible by 3.
108 is divisible by 3. So, we need to find such values for y that will make 5y divisible by 3.
Or in other words y should be a multiple of 3.
The values that y can take such that x does not become negative are 0, 3, 6, 9, 12, 15, 18, and 21.
So, there are 8 different combinations (36, 0), (31, 3), (26, 6), (21, 9), (16, 12), (11, 15), (6, 18) and (1, 21).
This question is a relatively easy question on LCM and HCF of two numbers.
If the product of two positive integers is 144, which of the following could be the LCM and HCF of the two numbers?
I. LCM : 24; HCF : 6
II. LCM : 18; HCF : 8
III. LCM : 16; HCF : 9
A. I only
B. II and III only
C. I and II only
D. I and III only
E. I, II and III
Correct Answer : Choice A
The product of two positive integers 'a' and 'b' is equal to the product of the LCM (a, b) and HCF (a, b).
i.e., a * b = LCM (a, b) * HCF (a, b)
HCF is a factor of both the positive integers.
LCM is a multiple of both the positive integers.
So, it is evident that the LCM of the two positive integers has to be a multiple of the HCF of the two numbers.
Combining these two results, we have to find out which among the pairs has a product of 144 such that the LCM is a multiple of the HCF.
I. LCM : 24 and HCF : 6. Product of the LCM and HCF = 24 * 6 = 144. The LCM 24 is a multiple of the HCF 6. Hence, this is a possible pair.
II. LCM : 18 and HCF : 8. Product of the LCM and HCF = 18 * 8 = 144. However, the LCM 18 is NOT a multiple of the HCF 8. Hence, this one is not a possible pair.
III. LCM : 16 and HCF : 9. Product of the LCM and HCF = 16 * 9 = 144. However, the LCM 16 is NOT a multiple of the HCF 9. Hence, this one is also not a possible pair.
Correct Answer is Choice C. Both statements together are sufficient to answer the question.
An "IS" question is answered when you can provide a definite YES or a definite NO as an answer to the question using the data.
We need to answer if a + b > 0.
Statement 1: a - b > 0.
We can infer that a > b.
If both a and b are negative and a > b, say a = -2 and b = -10, the sum of a and b, a + b < 0
On the contrary if both a and b are positive, the sum will be positive.
We cannot answer the question based on the data in statement 1.
Statement 1 is INSUFFICIENT.
Statement 2: |a| < |b|
The magnitude of a is less that of b.
a and b could both be negative. In that scenario a + b will be negative.
Both the numbers a and b could be positive. In that case a + b will be positive.
We cannot determine whether a + b is positive with this statement either.
Statement 2 is INSUFFICIENT.
Let us combine the data in the two statements.
a > b and |a| < |b|
If a and b are both positive, then if a > b, |a| also has to be greater than |b|.
i.e., for positive numbers larger the magnitude, larger the number.
So, we can infer that both a and b cannot be positive.
Either both a and b are negative or one is negative and the other is positive.
If both a and b are negative if a > b, |a| will be less than |b|. The sum of a and b, a + b < 0
If one of the two numbers is positive, a has to be positive as a > b.
If |a| is less than |b| as given in statement 2, then the magnitude of the positive number is lesser than the magnitude of the negative number.
So, the sum of a and b, a + b will be negative.
Hence, using the data in the two statements we can determine that a + b < 0.
So, the correct answer is choice C.
Here is an alternative method to determine this when combining the two statements.
Statement 1 : a - b > 0
Statement 2: |a| < |b|.
If |a| < |b|, we can conclude that a^2 < b^2.
So, we can determine that a^2 - b^2 < 0
a + b = (a^2 - b^2) / (a - b).
If a^2 - b^2 is negative and a - b is positive, a + b has to be negative.
Of the new vehicles registered in a week, 300 were neither SUVs nor were they powered by diesel. 3/4th of the SUVs registered were diesel powered and there were half as many SUVs as there were non SUVs. If SUVs not powered by diesel were a sixth of non SUVs not powered by diesel, how many vehicles registered in the week were powered by diesel? A. 200 B. 50 C. 250 D. 150 E. 400