Applied Mathematics · Class 12 Commerce
Ch 1Numbers, Quantification and Numerical Applications — Class 12 Applied Mathematics, concept-first.
This concept map shows how the whole chapter fits together. The topic Numbers, Quantification and Numerical Applications splits into two branches: Number Theory (Modulo Arithmetic, Congruence Modulo and Simple Arithmetic Functions) and Quantification & Numerical Application (Alligation and Mixture, Boats and Streams, P…
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Modular Arithmetic
You already know modular arithmetic. You just don't know you know it.
Most relevant Q&A
- Find (i) $21 \bmod 8$ (ii) $-31 \bmod 7$ (iii) $18 \bmod 18$ (iv) $7 \bmod 12$Free
- If $X = 7$ and $Y = 3$, then verify that $X \bmod Y = (X + kY) \bmod Y$ for $k = 6$.Free
- Verify that $(14 + 8) \bmod 5 = (14 \bmod 5 + 8 \bmod 5) \bmod 5$.Free
- Verify $(14 \times 8) \bmod 5 = (14 \bmod 5 \times 8 \bmod 5) \bmod 5$.Preview
- Find the remainder when $(127 \times 137 \times 23 \times 50 \times 235 \times 15)$ is divided by 7.Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Concept Map
This concept map shows how the whole chapter fits together. The topic Numbers, Quantification and Numerical Applications splits into two branches: Number Theory (Modulo Arithmetic, Congruence Modulo a…
Introduction
Numbers are often classified using a very simple test: what remainder do they leave when divided by another number? You already know this idea informally — classifying integers as even or odd is reall…
Modular Arithmetic
19 QModular arithmetic is the branch of arithmetic for integers that deals only with remainders, ignoring how many times one number divides into another.
+−Worked Examplesi9 questions
- Example 1Find (i) $21 \bmod 8$ (ii) $-31 \bmod 7$ (iii) $18 \bmod 18$ (iv) $7 \bmod 12$Free
- Example 2If $X = 7$ and $Y = 3$, then verify that $X \bmod Y = (X + kY) \bmod Y$ for $k = 6$.Free
- Example 3Verify that $(14 + 8) \bmod 5 = (14 \bmod 5 + 8 \bmod 5) \bmod 5$.Free
- Example 4Verify $(14 \times 8) \bmod 5 = (14 \bmod 5 \times 8 \bmod 5) \bmod 5$.Preview
- Example 5Find the remainder when $(127 \times 137 \times 23 \times 50 \times 235 \times 15)$ is divided by 7.Preview
- Example 6(i) Find addition modulo 8 if $a$ and $b$ are 3 and 11 respectively. (ii) Find addition modulo 25 where $a$ and $b$ are 15 and 6 respectivel…Preview
- Example 7(i) Find subtraction modulo 8 if $a$ and $b$ are 11 and 3 respectively. (ii) Apply subtraction modulo to positive integers 15 and 6 for modu…Preview
- Example 8(i) If $a = 3$, $b = 11$, $m = 8$, then find $a \odot_m b$. (ii) Apply multiplication modulo on positive integers 5 and 3 for modulo 25.Preview
- Example 9What time will it be after 1250 hours, if the present time is 9:00 pm?Preview
+−Exercise 1i10 questions
- Q1Find the sum of 132 and 121 mod 23.Free
- Q2Find $7^6 \pmod 3$.Free
- Q3Find the value of $x$, given that $x \equiv 23 \pmod 7$; if $21 \le x < 31$.Free
- Q4Find the remainder when $226 \times 369 \times 122 \times 461 \times 1025$ is divided by 8.Preview
- Q5Evaluate $(16 \times 29) \pmod 7$.Preview
- Q6If $x \pmod 9 = 2$, find all the possible values of $x$; where $0 < x < 47$.Preview
- Q7Find the positive integers less than 50 forming the equivalence class 4 for modulo 6.Preview
- Q8What time will it be after 200 hours, if the present time is 5:00 am?Preview
- Q9There are 81 boxes with 21 articles in each. When we rearrange all of the articles so that each box has 5 articles, how many articles will b…Preview
- Q10Find $3^{128} \bmod 7$.Preview
Application of Modulo Arithmetic
Modulo arithmetic is not just a classroom idea — because it captures the way many real systems "wrap around" after reaching a fixed size, it shows up across everyday life, mathematics, science and tec…
Congruence Modulo
Two positive integers and are said to be congruent modulo if they leave the same remainder when divided by .
+−Worked Examplesi4 questions
- Example 10Verify $a \equiv b \pmod m$, if $a = 41$, $b = 21$ and $m = 5$.Free
- Example 11If $a = 31$, $b = 21$, $m = 5$, then check $a \equiv b \pmod m$.Free
- Example 12Find $5^6 \pmod 4$.Preview
- Example 13Find all the positive integers less than 30 forming the equivalence class of 5 for modulo 7.Preview
Simple Arithmetic Functions
12 QRecall from earlier classes that a function is a rule that assigns to every element of one set (the domain) exactly one element of another set (the range) — a kind of machine that converts an input, o…
+−Worked Examplesi5 questions
+−Exercise 2i7 questions
- Q1Find $\varphi(35)$.Free
- Q2Find $\mu(75) + \mu(85)$.Free
- Q3Calculate $\varphi(90) + \tau(42) + \sigma(72) + \mu(70)$.Free
- Q4Verify that the relation $\varphi(p) + \tau(p) = \sigma(p)$ holds true for $p = 24$.Preview
- Q5If $p$ is a prime then $\varphi(p) + \tau(p) = \sigma(p)$. Using the information complete the following table: | $P$ | $\varphi(p)$ | $\tau(…Preview
- Q6If $n = 2p$ and $p$ is an odd prime number less than 14, then: (i) Evaluate $\varphi(n)$, $\tau(n)$, $\sigma(n)$ and represent the results i…Preview
- Q7Show that $\tau(n)$ and $\sigma(n)$ are multiplicative function for $n = 24$.Preview
Alligation and Mixture
10 QA mixture is formed by combining two or more ingredients together in a certain ratio. Alligation is the rule that works out exactly what ratio is needed — given the prices of two ingredients and the p…
+−Worked Examplesi2 questions
- Example 19A retailer has 250 kg of rice, a part of which he sells at 10% profit. The remaining quantity of rice is of low quality and he sold it at 5%…Free
- Example 20A container contains 40 litre milk. From this container 4 litre milk was taken out and replaced with water. This process was repeated furthe…Preview
+−Exercise 3i8 questions
- Q1In what ratio must rice at ₹69 per kg be mixed with rice ₹100 per kg so that the mixture be worth ₹80 per kg?Free
- Q2The average salary per head of the entire staff of a small factory including the supervisor and labours is ₹5750. The average salary per hea…Free
- Q3A container contains 70 l of orange squash. The squash being too concentrated 7 l of squash was taken out from this container and replaced b…Free
- Q4Cost of two types of pulses is ₹55 per kg and ₹90 per kg. If both the pulses are mixed together in the ratio 2:3, what should be the price o…Preview
- Q5A shopkeeper has 1 quintal of wheat, part of which she sells at 18% gain and the rest at 28% gain. In total she gains 24%. Find the quantity…Preview
- Q6600 gm of jaggery syrup has 40% jaggery in it. How much jaggery should be added to make it 50% in the syrup?Preview
- Q7In what ratio, water must be added to dilute honey costing ₹240 per litre so that the resulted syrup would be worth ₹200 per l?Preview
- Q8A container has 50 l of juice in it. 5 l of juice is taken out and is replaced by 5 l of water. This process is repeated 4 more times. What…Preview
Numerical Applications
This section brings together a set of practical numerical-reasoning problems that build directly on the modular arithmetic, congruence, and arithmetic-function ideas covered earlier in the chapter.
Boats and Streams
9 QBoats and streams problems deal with how the speed of a boat is affected by the current, or stream, it travels through.
+−Worked Examplesi2 questions
+−Exercise 4i7 questions
- Q1Find the speed of the boat, if a boat moves downstream at the rate of 16 km/hr and upstream at the rate of 10 km/hr.Free
- Q2The speed of a boat in still water is 14 km per hour. While going downstream it moves at the rate of 24 km per hour. Find the speed of the b…Free
- Q3A boat goes 8 km upstream and then returns. Total time taken is 4 hours 16 minutes. If the speed of current is 1 km/hr, find the actual spee…Free
- Q4A man can row 7 km per hour in still water. If the stream is flowing at the rate of 5 km per hour, it takes him 7 hours to row to a place an…Preview
- Q5A boat covers 32 km upstream and 36 km downstream in 7 hours. Also it covers 40 km upstream and 48 km downstream in 9 hours. Find the speed…Preview
- Q6A man can row $7\frac{1}{2}$ km/h in still water. If in a river running at 1.5 km an hour, it takes him 50 minutes to row to a place and bac…Preview
- Q7The speed of a motor boat and that of the current of water is 36:5. The boat goes along with the current in 5 hours 10 minutes. How much tim…Preview
Pipes and Cisterns
9 QPipes and cisterns problems apply the idea of rates to tanks being filled or drained. A pipe that fills a tank is called an inlet pipe, while one that drains or empties it is called an outlet pipe.
+−Worked Examplesi2 questions
- Example 23Two pipes can fill a cistern in 8 and 12 hours respectively. The pipes are opened simultaneously and it takes 12 minutes more to fill the ci…Free
- Example 24A cistern can be filled by two pipes $A$ and $B$ in 12 minutes and 15 minutes respectively. Another tap $C$ can empty the full tank in 20 mi…Preview
+−Exercise 5i7 questions
- Q1Pipe $A$ can fill a tank in 30 hours and pipe $B$ in 45 hours. If both the pipes are opened in an empty tank, how much time will it take to…Free
- Q2A pipe can fill a cistern in 6 hours. Due to a leakage in the tank the cistern is just full in 9 hours. How much time the leakage will take…Free
- Q3A cistern can be filled by pipes $A$ and $B$ in 4 hours and 6 hours respectively. When full, the cistern can be emptied by pipe $C$ in 8 hou…Free
- Q4A cistern can be filled in 8 hours but due to a leakage in its bottom, it takes 2 hours more to fill the tank. If the cistern is full, how m…Preview
- Q5A cistern can be filled by an inlet pipe in 20 hours and can be emptied by an outlet pipe in 25 hours. Both the pipes are opened. After 10 h…Preview
- Q6Two pipes $A$ and $B$ can fill a tank in 24 minutes and 32 minutes respectively. If both the pipes are opened simultaneously, after how much…Preview
- Q7A tank is fitted with 3 taps $A$, $B$ and $C$. All the three taps, if opened together, can drain the full tank in $1\frac{1}{2}$ minutes. Ta…Preview
Races and Games
10 QA race is a contest in which competitors cover a fixed distance in the shortest possible time — running, riding, driving, sailing, and rowing are all common forms.
+−Worked Examplesi3 questions
- Example 25In a 1000 metres race, $A$, $B$ and $C$ get the gold, silver and bronze medals, respectively. If $A$ beats $B$ by 100 metres and $B$ beats $…Free
- Example 26Priya can run 1 km in 3 min 10 sec and Preeti can cover the same distance in 3 min 20 sec. By what distance can Priya beat Preeti?Preview
- Example 27Two runners $A$ and $B$ complete a 100 metres race in 36 seconds and 48 seconds respectively, by how many metres will $A$ defeat $B$?Preview
+−Exercise 6i7 questions
- Q1In a 1000 metres race, $A$ defeats $B$ by 300 metres and $B$ defeats $C$ by 200 metres. In the same race by how many metres will $A$ defeat…Free
- Q2In a 1000 metres race, $A$ can give a start of 100 metres to $B$ and a start of 280 metres to $C$. In the same race, how much start can $B$…Free
- Q3In a 500 metres race, $A$ defeats $B$ by 60 metres (or) 12 seconds. What is the time taken by $A$ to complete the race?Free
- Q4In a 900 metres race, $A$ gives $B$ a start of 150 metres and defeats him by 50 seconds. If the speed of $A$ is 4.5 m/sec then find the spee…Preview
- Q5$A$ runs 3 times as fast as $B$. If $A$ gives $B$ a start of 40 metres, how far must the goal on the race course be so that $A$ and $B$ reac…Preview
- Q6A team played 40 games in a season and lost in 16 of them. What percent of games played did the team win?Preview
- Q7In a 200 metres race, Prateek beats Samarth by 35 metres or 7 seconds. How much time did Prateek take to cover the race?Preview
Partnership
11 QA partnership is formed when two or more people agree to jointly run a business as co-owners, sharing in its risks, responsibilities, and — most importantly for this topic — its profits and losses.
+−Worked Examplesi3 questions
- Example 28$P$, $Q$ and $R$ jointly invested a total of ₹64000 in a business. $P$ invested ₹6000 more than $Q$ and $Q$ invested ₹5000 more than $R$. Fi…Free
- Example 29Divide a net profit of ₹21000 among three partners $A, B, C$ in a business whose capital investments were ₹36000, ₹42000 and ₹48000 respecti…Preview
- Example 30Three friends $A, B$ and $C$ enter into a partnership to run a café business. $A$ puts in ₹5000 per month for the whole year, $B$ contribute…Preview
+−Exercise 7i8 questions
- Q1$A, B$ and $C$ enter into a partnership. $B$ contributes $\frac{1}{3}$ of the capital, while $A$ contributes as much as $B$ and $C$ together…Free
- Q2Rahul got ₹5000 as his share out of the total profit of ₹9000. Ramesh had invested ₹3000 for 6 months, while Rahul invested for the whole ye…Free
- Q3Priya started a tiffin service business by investing ₹40000. After 4 months her friend Rekha joined her business and invested ₹50000 in it.…Free
- Q4What is the investment ratio of $A, B$ and $C$ if their profit ratio is 6:7:8 respectively and their investment time period ratio is 2:3:4 r…Preview
- Q5$A$ and $B$ start a trading business with a capital of ₹30000 and ₹60000 respectively. From the profit of entire year which is ₹18000, $A$ g…Preview
- Q6Prakash, Sachin and Anil started a business jointly investing ₹11 lakh, ₹16.5 lakh and ₹8.25 lakh respectively. The profit earned by them in…Preview
- Q7Nikhil, Priyesh and Ritik took a house on rent for ₹13824. They remained together for 4 months and then Ritik left the house. After 5 more m…Preview
- Q8Arun started a business investing ₹38000. After 5 months Bakul joined him with a capital of ₹55000. At the end of the year the total profit…Preview
Scheduling
10 QScheduling is the art of planning activities to achieve a goal within limited time and resources. It shows up anywhere multiple tasks compete for the same limited capacity — from coordinating a large…
+−Worked Examplesi4 questions
- Example 31The processes P1, P2, P3, P4 given in the table below, arrives for execution in the same order, with Arrival Time 0, and given Burst Time. F…Free
- Example 32Consider the available processes given below in the ready queue for execution and with given burst time. | Process No | Arrival Time | Burst…Free
- Example 33Consider the below processes available in the ready queue for execution, with arrival time as 0 for all and given burst time. Find the avera…Preview
- Example 34Consider the below processes available in the ready queue for execution and with given burst time. | Process No | Arrival Time | Burst time…Preview
+−Exercise 8i6 questions
- Q1Which of the following is false about SJF? S1: It causes minimum average waiting time S2: It can cause starvation (A) Only S1 (B) Only S2 (C…Free
- Q2What is the relationship between arrival time, burst time, completion time and waiting time?Free
- Q3The processes P1, P2, P3, P4 given in the table below, arrives for execution in the same order, with Arrival Time 0, and given Burst Time, f…Preview
- Q4Consider the available processes given below in the ready queue for execution, with arrival time as 0 for all and given burst time. Find the…Preview
- Q5Consider the available processes given below in the ready queue for execution and with given burst time. | Process No | Arrival Time | Burst…Preview
- Q6Consider the available processes given below in the ready queue for execution and with given burst time. | Process No | Arrival Time | Burst…Preview
Numerical Inequalities
12 QAn inequality is a statement that compares two quantities and shows that one is less than, greater than, or otherwise not equal to the other, using the symbols , , , or .
+−Worked Examplesi2 questions
+−Exercise 9i10 questions
- Q1Mr. X and Mr. Y have net worth of ₹1.93 crores INR and $-0.22$ crores INR. Represent the above information in the form of inequality.Free
- Q2Two players: Player A and Player B are playing a game by rolling a dice. They decided that the player who will get the higher total will be…Free
- Q3Solve: $4x - 2 < 8$, when $x \in \mathbb{Z}$.Free
- Q4Show that the numbers 16 and 4, satisfy the numerical inequality $AM \ge GM$.Preview
- Q5Solve the following inequality: (i) $(-2z - 6) < 10$ (ii) $2a < a - 4 \le 3a + 8$ (iii) $\dfrac{(y-1)}{3} + 4 < \dfrac{(y-5)}{5} - 2$Preview
- Q6Prove that the following inequality holds true: $\sqrt{5} + \sqrt{3} > \sqrt{6} + \sqrt{2}$Preview
- Q7Satyarth and Swarit are brothers. Satyarth owns a house which is worth ₹3 crores and Swarit owns a farmhouse which is worth ₹2.75 crores. Bu…Preview
- Q8Insert the appropriate sign of inequality: $\sqrt{3}(\sqrt{50} - \sqrt{32}) \underline{\quad\quad} 3\sqrt{54} + 2\sqrt{24}$Preview
- Q9If $a$ and $b$ are positive integers and $\dfrac{a-b}{6.25} = \dfrac{8}{2.5}$ (i) $b > a$ (ii) $b < a$ (iii) $b = a$ (iv) $b \ge a$Preview
- Q10If $p > q$ and $r < 0$, then which of the following is true? (i) $pr < qr$ (ii) $p - r < q - r$ (iii) $p + r < q + r$ (iv) None of thesePreview