Question 1

The price of a mobile phone is ₹8,250. A GST of 18% is added to the price. Which of the following gives the final price of the phone including the GST?

(i) 8250 + 18
(ii) 8250 + 1800
(iii) 8250 + \frac{18}{100}
(iv) 8250 × 18
(v) 8250 × 1.18
(vi) 8250 + 8250 × 0.18
(vii) 1.8 × 8250

Accepted Answer

The GST is 18% of ₹8250, which is (18/100) × 8250 = ₹1485. The final price is the original price plus GST, i.e., 8250 + 1485 = ₹9735.

Among the options, (v) 8250 × 1.18 = 8250 × (1 + 0.18) = 9735 and (vi) 8250 + 8250 × 0.18 = 8250 + 1485 = 9735 both correctly represent the final price including GST.

Option (ii) 8250 + 1800 is incorrect because 1800 is not 18% of 8250.

Option (vii) 1.8 × 8250 = 14850 is incorrect (it multiplies by 180%, not 118%).

Therefore, correct expressions are (v) and (vi). — GST is calculated as 18% of the price: (18/100) × 8250 = 1485.
Final price = original price + GST = 8250 + 1485 = 9735.

Expressions that correctly calculate this are those that multiply 8250 by 1.18 or add 18% of 8250 to 8250.

Hence, options (v) and (vi) are correct.

Question 2

The monthly percentage change in population (compared to the previous month) of mice in a lab is given: Month 1 change was +5%, Month 2 change was -2%, and Month 3 change was -3%. Which of the following statement(s) are true? The initial population is p.

(i) The population after three months was p × 0.05 × 0.02 × 0.03.
(ii) The population after three months was p × 1.05 × 0.98 × 0.97.
(iii) The population after three months was p + 0.05 - 0.02 - 0.03.
(iv) The population after three months was p.
(v) The population after three months was more than p.
(vi) The population after three months was less than p.

Accepted Answer

The population changes monthly by +5%, -2%, and -3% respectively.

This means the population after Month 1 = p × (1 + 5/100) = p × 1.05
After Month 2 = previous × (1 - 2/100) = p × 1.05 × 0.98
After Month 3 = previous × (1 - 3/100) = p × 1.05 × 0.98 × 0.97

(i) is incorrect because 0.05, 0.02, 0.03 are decimal forms of 5%, 2%, 3%, but population changes multiply by (1 + change), not by the change itself.

(iii) is incorrect because population changes are multiplicative, not additive.

(iv) is incorrect because population changes.

(v) or (vi) depends on the product 1.05 × 0.98 × 0.97 = approx 1.05 × 0.9506 = approx 0.998, which is less than 1, so population after 3 months is slightly less than p.

Therefore, correct statements are (ii) and (vi). — Population changes by percentages are multiplicative factors: (1 + change/100).

Calculate: 1.05 × 0.98 × 0.97 ≈ 0.998, which is less than 1.

Hence, population after 3 months is slightly less than initial p.

Statements (ii) and (vi) are true.

Question 3

A shopkeeper initially set the price of a product with a 35% profit margin. Due to poor sales, he decided to offer a 30% discount on the selling price. Will he make a profit or a loss? Give reasons for your answer.

Accepted Answer

Let the cost price (CP) be ₹100.

Profit margin is 35%, so selling price (SP) = 100 + 35 = ₹135.

Now, a 30% discount is offered on SP, so new selling price = 135 - 30% of 135 = 135 - 40.5 = ₹94.5.

Since new SP (₹94.5) < CP (₹100), the shopkeeper makes a loss.

Loss = CP - SP = 100 - 94.5 = ₹5.5.

Loss percentage = (5.5 / 100) × 100% = 5.5%. — Calculate selling price with profit margin.
Apply discount on selling price.
Compare discounted price with cost price.
If discounted price < cost price, loss occurs.
Calculate loss and loss percentage accordingly.

Question 4

What percentage of area is occupied by the region marked 'E' in the figure?

Accepted Answer

The question refers to a figure (not provided here). To find the percentage area occupied by region 'E', calculate the area of 'E' and divide by the total area, then multiply by 100.

Without the figure, the exact numerical answer cannot be provided. — Identify the total area and the area of region 'E' from the figure.
Calculate (Area of E / Total Area) × 100 to get the percentage.

Since figure is not provided, solution cannot be completed here.

Question 5

What is 5% of 40? What is 40% of 5?
What is 25% of 12? What is 12% of 25?
What is 15% of 60? What is 60% of 15?
What do you notice?
Can you make a general statement and justify it using algebra, comparing x% of y and y% of x?

Accepted Answer

Calculate each:
5% of 40 = (5/100) × 40 = 2
40% of 5 = (40/100) × 5 = 2
25% of 12 = (25/100) × 12 = 3
12% of 25 = (12/100) × 25 = 3
15% of 60 = (15/100) × 60 = 9
60% of 15 = (60/100) × 15 = 9

Observation: x% of y = y% of x.

General statement: For any numbers x and y,

x% of y = (x/100) × y = (y/100) × x = y% of x.

This shows the two expressions are equal. — Calculate each pair to see they are equal.
Use algebra to show x% of y = y% of x.
This is because multiplication is commutative.

Question 6

A school is organising an excursion for its students. 40% of them are Grade 8 students and the rest are Grade 9 students. Among these Grade 8 students, 60% are girls. [Hint: Drawing a rough diagram can help].

(i) What percentage of the students going to the excursion are Grade 8 girls?
(ii) If the total number of students going to the excursion is 160, how many of them are Grade 8 girls?

Accepted Answer

(i) Percentage of Grade 8 students = 40%
Among Grade 8 students, girls = 60%

Percentage of Grade 8 girls = 40% × 60% = (40/100) × (60/100) = 24%

(ii) Total students = 160
Number of Grade 8 girls = 24% of 160 = (24/100) × 160 = 38.4 ≈ 38 students (if rounding down) or 39 (if rounding up).

Usually, number of students is whole number, so 38 or 39 girls. — Multiply percentages to find combined percentage.
Apply percentage to total number to find count.
Use rounding as appropriate.

Question 7

A shopkeeper sells pencils at a price such that the selling price of 3 pencils is equal to the cost of 5 pencils. Does he make a profit or a loss? What is his profit or loss percentage?

Accepted Answer

Let cost price (CP) of one pencil = ₹x.

Cost price of 5 pencils = 5x.
Selling price (SP) of 3 pencils = 5x (given).

Therefore, SP of one pencil = (5x)/3.

Profit per pencil = SP - CP = (5x/3) - x = (5x - 3x)/3 = (2x)/3.

Profit percentage = (Profit / CP) × 100 = [(2x/3) / x] × 100 = (2/3) × 100 = 66.67% profit. — Compare cost price and selling price per pencil.
Calculate profit per pencil.
Calculate profit percentage using formula.

Question 8

The bus fares were increased by 3% last year and by 4% this year. What is the overall percentage price increase in the last 2 years?

Accepted Answer

Let the initial fare be ₹100.

After 3% increase last year: 100 × 1.03 = ₹103.

After 4% increase this year: 103 × 1.04 = ₹107.12.

Overall increase = 107.12 - 100 = ₹7.12.

Overall percentage increase = (7.12 / 100) × 100% = 7.12%. — Calculate successive increases by multiplying factors.
Subtract initial amount to find increase.
Calculate percentage increase accordingly.

Question 9

If the length of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) does the breadth decrease by?

Accepted Answer

Let original length = L and breadth = B.

Original area = L × B.

Length increased by 10%: new length = 1.10 × L.

Area unchanged, so new area = original area.

New area = new length × new breadth = 1.10 L × new breadth = L × B.

Therefore, new breadth = (L × B) / (1.10 L) = B / 1.10 = (10/11) B.

Breadth decreases from B to (10/11) B.

Percentage decrease = [(B - (10/11) B) / B] × 100 = (1 - 10/11) × 100 = (1/11) × 100 ≈ 9.09%. — Use formula for area.
Set new area equal to original area.
Solve for new breadth.
Calculate percentage decrease accordingly.

Question 10

The percentage of ingredients in a 65 g chips packet is shown in the picture. Find out the weight each ingredient makes up in this packet.

Accepted Answer

Assuming the percentages of ingredients are given in the figure (not provided here),

Weight of each ingredient = (percentage / 100) × 65 g.

Calculate for each ingredient accordingly.

Exact weights cannot be provided without the figure. — Multiply total weight by percentage (as decimal) to find weight of each ingredient.

Without figure, exact calculation is not possible.

Question 11

Three shops sell the same items at the same price. The shops offer deals as follows:

Shop A: "Buy 1 and get 1 free"
Shop B: "Buy 2 and get 1 free"
Shop C: "Buy 3 and get 1 free"

Answer the following:

(i) If the price of one item is ₹100, what is the effective price per item in each shop? Arrange the shops from cheapest to costliest.
(ii) For each shop, calculate the percentage discount on the items. [Hint: Compare the free items to the total items you receive.]
(iii) Suppose you need 4 items. Which shop would you choose? Why?

Accepted Answer

(i) Price per item = ₹100.

Shop A: Buy 1 get 1 free means pay for 1 item and get 2 items.
Effective price per item = 100 / 2 = ₹50.

Shop B: Buy 2 get 1 free means pay for 2 items and get 3 items.
Effective price per item = (2 × 100) / 3 ≈ ₹66.67.

Shop C: Buy 3 get 1 free means pay for 3 items and get 4 items.
Effective price per item = (3 × 100) / 4 = ₹75.

Order from cheapest to costliest: Shop A (₹50), Shop B (₹66.67), Shop C (₹75).

(ii) Percentage discount = (Number of free items / Total items) × 100.

Shop A: 1 free out of 2 items = (1/2) × 100 = 50% discount.
Shop B: 1 free out of 3 items = (1/3) × 100 ≈ 33.33% discount.
Shop C: 1 free out of 4 items = (1/4) × 100 = 25% discount.

(iii) Need 4 items.

Shop A: Buy 2 items to get 2 free, total 4 items, pay 2 × 100 = ₹200.
Shop B: Buy 3 items to get 1 free, total 4 items, pay 3 × 100 = ₹300.
Shop C: Buy 3 items to get 1 free, total 4 items, pay 3 × 100 = ₹300.

Choose Shop A to pay least amount (₹200) for 4 items. — Calculate effective price per item by dividing total price paid by total items received.
Calculate discount percentage by comparing free items to total items.
For 4 items, calculate total cost in each shop and choose the least.

Question 12

In a room of 100 people, 99% are left-handed. How many left-handed people have to leave the room to bring that percentage down to 98%?

Accepted Answer

Total people = 100
Left-handed people = 99

Let x be the number of left-handed people who leave.

After x leave, left-handed people = 99 - x
Total people = 100 - x

We want percentage left-handed = 98%

So, (99 - x) / (100 - x) = 98 / 100 = 0.98

Cross multiply:
100(99 - x) = 98(100 - x)
9900 - 100x = 9800 - 98x

Bring variables to one side:
9900 - 9800 = 100x - 98x
100 = 2x
x = 50

Therefore, 50 left-handed people must leave to reduce percentage to 98%. — Set up equation using percentage formula.
Solve for x.
Interpret result.

Question 13

Look at the following graph.

Ability to use computer by age and gender (2023)

Based on the graph, which of the following statement(s) are valid?

(i) People in their twenties are the most computer-literate among all age groups.
(ii) Women lag behind in the ability to use computers across age groups.
(iii) There are more people in their twenties than teenagers.
(iv) More than a quarter of people in their thirties can use computers.
(v) Less than 1 in 10 aged 60 and above can use computers.
(vi) Half of the people in their twenties can use computers.

Accepted Answer

Based on the graph:

(i) True - The graph shows highest computer literacy in twenties.
(ii) True - Women have lower percentages across age groups.
(iii) Cannot be determined from graph (graph shows literacy %, not population count).
(iv) True - More than 25% (a quarter) in thirties can use computers.
(v) True - Less than 10% (1 in 10) aged 60+ can use computers.
(vi) False - The graph shows about 40-45%, not exactly half.

Therefore, statements (i), (ii), (iv), and (v) are valid. — Interpret the graph carefully.
Check each statement against data shown.
Note difference between percentage and population count.

Question 14

What is a fraction and how is it generally represented in mathematics?

Accepted Answer

A fraction is a number that represents parts of a whole. It is generally represented as a/b, where 'a' is the numerator indicating the number of parts considered, and 'b' is the denominator indicating into how many equal parts the whole is divided. — A fraction is a way to express a part of a whole quantity. For example, in the fraction 3/4, 3 is the numerator showing how many parts are taken, and 4 is the denominator showing the total equal parts the whole is divided into.

Question 15

Two fractions are given as $\frac{2}{3}$ and $\frac{4}{6}$. Are these fractions equivalent? Justify your answer using cross multiplication.

Accepted Answer

Yes, the fractions $\frac{2}{3}$ and $\frac{4}{6}$ are equivalent. Using cross multiplication, 2 × 6 = 12 and 3 × 4 = 12. Since both products are equal, the fractions are equivalent. — Cross multiplication helps to check equivalence of fractions. For $\frac{2}{3}$ and $\frac{4}{6}$, multiplying numerator of first by denominator of second gives 12, and denominator of first by numerator of second also gives 12. Equal products mean the fractions represent the same value.

Disguise

Disguise — Study Notes

Introduction

Fractions and Their Equivalence

Decimals and Their Relation to Fractions

Practice Questions — Disguise

All 7 Chapters in Ganita Prakash Part-II