If a, b, c and g are non-zero integers, 3 ≥ c ≥ – 9 , 5 ≥ a ≥ –2 , 10

This question tests you on the max/min concept of inequalities.

Let us first discuss the when and how to use the max/min concept of inequalities:

When to use the Max/Min Concept of Inequalities:

Whenever you encounter a question with two finite ranges (x and y in this case) and the question asks us to find the sum (x+y), difference (x-y) and product (xy) of the two ranges, then this concept needs to be used

How to use the Max/Min Concept of Inequalities:

1. Place the two finite ranges one below the other
2. Make sure the inequality signs are the same. If they are not the same then we make them the same by flipping one finite ranges inequality sign. This can be done by reversing the inequality or multiplying throughout by -1
3. Perform the mathematical operation only between the extreme values of the finite ranges.

The question here clearly has two finite ranges and asks us information about the product xy, so we can definitely use the Max/Min concept for both the numerator and the denominator.

5 ≥ a ≥ –2
4 ≥ g ≥ 1

The 4 products we get for ag are 20, -2, 5, -8. Taking the greatest and the least values we get 20 ≥ ag ≥ -8.

10 ≥ b ≥ –1
3 ≥ c ≥ – 9

The 4 products we get for bc are 30, 9, -90 and -3. Taking the greatest and the least values we get 30 ≥ bc ≥ -90

Now since we require the smallest possible value of ag/bc, we need the denominator to have a smaller magnitude than the numerator and the sign of ag/bc needs to be negative.

So ag = 20 and bc = -1 ----> ag/bc = 20/-1 = -20

Answer : B

Takeaway : The max/min concept can be used to quickly find the range of a sum, difference or a product of values. When divided two values, the max/min concept cannot be used and the question must be solved using numbers for the numerator and denominator.

Hope this helps!

Aditya
CrackVerbal Quant Expert
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smallest value is when numerator is greater ag = 5*4 = 20
denominator is greatest negative value bc = -1*1 = -1

= -20

Ans B

Because we want the smallest possible value of ag/bc, we can first see that the value of the fraction will have to be negative. Thus, either the product ag will be negative, or bc will be negative. Additionally, we want the absolute value of the entire fraction to be as large as possible, but still in keeping with the given constraints.

Let’s start with the product ag:

We are given that 5 ≥ a ≥ –2 and 4 ≥ g ≥ 1

If we let a = 5 and g = 4, we have ag = 20.

Let’s now move to the denominator bc:

We are given that 3 ≥ c ≥ – 9 and 10 ≥ b ≥ –1

If we let c = 1 and b = -1, we have bc = -1.

Thus, the smallest value of ag/bc = 20/-1 = -20.

Answer: B
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C = -9 to 3 and B = -1 to 10
A = -2 to 5 and G=1 to 4

AG/BC will be minimum when the number is negative and has denominator 1; so Pick 5 and 4 for AG; -1 and 1 for B and C so -20 is the lowest IMO B

Hi everyone,

If a, b, c and g are non-zero integers, 3 ≥ c ≥ – 9 , 5 ≥ a ≥ –2 , 10 ≥b ≥ –1 , 4 ≥ g ≥ 1, then which of the following expresses the smallest possible value of ag/(bc)?

Pre-thinking:
Basically we have a fraction and we are looking for combining numbers that yield the minimum value possible.
Note that we have an array of choices that gives us the opportunity to have negative results.
So we are looking for two things:
Nominator: the highest positive value
Denominator: the lowest negative value.

Note that this approach depends and the array of numbers that we have.

Nominator=5*4=20
Denominator=-1*1=-1




(A) –90
(B) –20
(C) –10
(D) 1/90
(E) 15/4

As we can observe that ag/(bc) assumes negative values (for instance, we can take a = -1 and b = c = g = 1 to have ag/(bc) = -1), we can eliminate D and E and focus on negative values.

Since a, b, c and g are integers and since the smallest value of ag/(bc) is negative, the denominator of ag/(bc) should be 1 or -1. This is because for a negative number, any integer value of denominator other than 1 or -1 will result in a greater number (for instance, if we have the number -5, if we change the denominator from 1 to 2, we will get -5/2 = -2.5; which is greater than -5).

If the denominator is 1, then the numerator should be negative and since g is between 1 and 4 i.e. positive, “a” has to be negative. In this case, the smallest number we can form is -8; which corresponds to choosing a = -2, g = 4, b = c = 1 (or b = c = -1).

If the denominator is -1, then the numerator should be positive and as large as possible. To obtain the greatest absolute value, we should pick the greatest possible values for a and g, which are 5 and 4, respectively. In this case, we obtain -20; which corresponds to choosing a = 5, g = 4, b = 1 and c = -1 (or b = -1 and c = 1). As we can see, the smallest value for the expression ag/(bc) is -20.

Answer: B
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im really confused by the answer

3 ≥ c ≥ – 9
5 ≥ a ≥ –2
10 ≥ b ≥ –1
4 ≥ g ≥ 1
a=1
g=1
b=10
c=-9
ag/bc = (1)(1)/(10)(-9)

-1/90 should be the answer..why is this wrong?

edit: NVM.. OFC -20 is smaller than -1/90

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