# FREE IMO Logical Reasoning Questions and Answers

#### If the reflection of a clock in the mirror shows the time as 4:35, find the actual time.

Explanation:

When you look at a reflection in a mirror, the hour and minute hands appear to be switched. So, if the mirror shows 4:35, it means that the actual time is 7:25.

#### Pradip and Meenakshi are racing on a circular track. If Pradip takes 40 minutes and Meenakshi takes 35 minutes to complete the round. If they both start at the same point at the same time and go in the same direction, after how many minutes will they meet again at the start point?

Explanation:

The least common multiple (LCM) of 40 minutes and 35 minutes is 280 minutes. This is when they will meet again at the start point.

#### Harsh plays hockey every 4th day and plays squash every 6th day. He played both hockey and squash on 16th June. What is the next date when Harsh will play both games?

Explanation:

Harsh plays hockey every 4th day and squash every 6th day. The least common multiple (LCM) of 4 and 6 is 12. So, Harsh will play both hockey and squash every 12 days. Since he played both games on 16th June, the next date when he will play both games is 16 + 12 = 28th June.

#### Determine the two numbers nearest to 9000 which are exactly divisible by 4, 9, 6, and 5.

Explanation:

To find numbers divisible by 4, 9, 6, and 5 nearest to 9000, we need to find the least common multiple (LCM) of these numbers, which is 180.

The nearest numbers divisible by 180 around 9000 are 9000 and 9180.

#### Aman is standing in a row. If he is 12th from the left and 16th from the right, how many people are standing in the row?

Explanation:

To find the total number of people standing in the row, we'll add Aman's position from the left and the right and then subtract 1 because Aman is counted twice in the sum.

So, 12+16−1=27

#### Ria is standing in a row. If she is 6th from the left and 20th from the right, how many people are standing in the row?

Explanation:

Ria is 6th from the left and 20th from the right. To find the total number of people standing in the row, we add Ria's position from the left and the right, and then subtract 1 because Ria is counted twice in the sum.

So, 6 + 20 − 1 = 25

#### A rectangular courtyard with a length of 3 m 25 cm and a breadth of 2 m 35 cm is to be paved with square stones of the same size. Find the least number of such stones required.

Explanation:

To find the least number of square stones required, divide the area of the rectangular courtyard by the area of one square stone. The dimensions of the courtyard are 325 cm by 235 cm, so its area is 76,375 square centimeters. Assuming each stone is x cm by x cm, we want to maximize x. The greatest common divisor (GCD) of 325 and 235 is 5, so each stone is 5 cm by 5 cm, covering 25 square centimeters. Dividing the area of the courtyard by the area of one stone gives us 3,055 stones.

#### Divya plays squash every 3rd day and plays football every 4th day. She played both squash and football on 4th April. What is the next date when Divya will play both games?

Explanation:

Divya plays squash every 3rd day and football every 4th day. The next date when she plays both games will be the least common multiple (LCM) of 3 and 4, which is 12. Adding 12 days to April 4th, we get April 16th, when Divya will play both squash and football again.

#### The product of the two numbers is 108. If their difference is 23, find the numbers.

Explanation:

Let the numbers be x and y. We have two equations: xy = 108 and x - y = 23. Solving these equations gives us x = 27 and y = 4.

#### A lighthouse has two lights - one that flashes every 4 minutes and another that flashes every 3½ minutes. Suppose the lights flash together at noon. What is the first time after 2 PM when they will flash together again?

Explanation:

The least common multiple (LCM) of 4 minutes and 3½ minutes is 7 minutes. Adding 7 minutes to 2:00 PM gives us 2:07 PM. Since 2:07 PM is not an option, the next time after 2:07 PM is 2:14 PM, but since the light flashes together at noon, the next time they flash together is 2:20 PM.

#### Showing the man singing on the stage, Sarika says, "He is the brother of my uncle's daughter". How is Sarika related to the man?

Explanation:

Sarika's uncle's daughter is her cousin. The man singing on the stage is stated to be the brother of Sarika's cousin, so he is also Sarika's cousin.