Embibe Experts Solutions for Chapter: Real Numbers, Exercise 1: Exercise

Ajay always goes to his favourite barber shop every $4^{\mathrm{th}}$ day for styling his beard. Wednesday is a very busy day for Ajay as he has a lot of meetings on Wednesday, so he never wants to go to the barber shop on Wednesday. But this week, due to some urgency, he has to go to the barber shop on Wednesday itself.

If he goes to the barbershop this Wednesday, then how many days later will he be visiting the barber shop again on a Wednesday?

Non-terminating and repeating decimal are represented by putting a bar on the repeated part. For example: $a.bcbcbcbcbcb.....$ can be represented as $a.\overline{bc}$. 

Angad went to a shop to buy a toffee. When he asked about the price of toffee to the shopkeeper, the shopkeeper told him the cost of the toffee is $₹1.325325. . . .$ , he wants to convert the cost into a fractional form. Help Angad to convert the cost of toffee into a fractional form.

In today’s class, Arjun and Rahul studied Euclid’s Division Lemma and Euclid's division algorithm.

Euclid's division algorithm states that for any two given positive integers $a$ and $b$, there exist unique integers $q$ and $r$ satisfying $a=bq+r, 0\le r<b$.

Arjun asked Rahul to find the H.C.F of $1260$ and $7344$ using Euclid's division algorithm. Rahul found the answer as $36$.

Is Rahul correct? Show the process Rahul used to get the answer.

Raj is playing "Roll a dice-Number game" with his friend Rahul. If he gets a $5$ on the dice, he is supposed to collect an Irrational number from Rahul. Rahul has the set of numbers: $\left\{3, 3.14, \sqrt{2},\frac{1}{2}\right\}$ with him. Raj takes the turn and rolls the dice. He gets a $5$ on the dice, so he picks up the number $\sqrt{2}$ from Rahul. Simran, who was just observing the game, said that the number collected by Raj is not an Irrational number. Help Raj to prove that the number collected by him from Rahul is an Irrational number.

A milk processing company has two different suppliers delivering milk to them. One of them has all their jars of $250$ litres capacity. The other one has jars of capacity $425$ litres. The Operations department of the company now wants to minimise cost, and wants to buy only one size of container which can measure both the jar capacities in complete multiples. What is the maximum capacity of the container that the Operations department should get?

Indian army contingent of $616$ members are to march behind the Indian army band of $32$ members in republic day parade of $2023$ at Raj Path.

The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

Parul and Nupur both study in class $\mathrm{X}$. Today, they were taught about the Fundamental Theorem of Arithmetic which states that, "Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur".

To explain the Fundamental Theorem of Arithmetic, their teacher recalled the prime factorisation of a number by the 'factor tree method'. 

Parul became interested in the factor tree method and she asked Nupur to help to complete the following factor tree.

Find the missing numbers.

Once upon a time there used to be a mythical city named El Dorado (The City of Gold). A merchant who claims to know the location of this city, has arrived at the port of Mumbai. He has brought with him $120$ litres of liquid gold of one kind, $180$ litres of liquid gold of another kind and $240$ litres of the liquid gold of the third kind. He wants to sell the gold to the biggest gold smuggler of the city, by filling the three kinds of golds in tins of equal capacity. What should be the greatest capacity of such a tin?