Another Peek Beyond the PointClass 7 Mathematics NCERT Solutions
35 Solutions
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Solution 1 of 35
Q1Figure it Out (Chapter End)
A 210 gram packet of peanut chikki costs ₹70.5, while a 110 gram packet of potato chips costs ₹33.25. Which is cheaper?
Solution
Given:
Peanut chikki: 210 g for ₹70.5
Potato chips: 110 g for ₹33.25
To Find:
Which item is cheaper per gram.
Solution:
To compare, we need to find the cost per gram for each item.
Cost per gram of peanut chikki:
Cost per gram of potato chips:
Comparison:
The cost per gram of potato chips is less than the cost per gram of peanut chikki.
Final Answer: The potato chips are cheaper.
Q2Figure it Out (Chapter End)
Write the decimal number at the arrow mark:
Solution
Description of the image:
The question shows a number line segment from 0.7 to 0.8. This segment is divided into 10 equal smaller intervals. The arrow points to the 4th mark after 0.7.
Solution:
Each small interval represents one-tenth of the distance between 0.7 and 0.8. The total distance is .
So, each small interval represents .
The marks represent 0.71, 0.72, 0.73, 0.74, and so on.
The arrow is pointing at the 4th mark, which corresponds to the number 0.74.
Final Answer: 0.74
Q3Figure it Out (Chapter End)
Shyamala bought 3 kg bananas at ₹30/- per kg. She counted 35 bananas in all. She sells each banana for ₹5/-. How much profit does she make selling all the bananas?
Solution
Given:
Quantity of bananas bought = 3 kg
Cost of bananas = ₹30 per kg
Total number of bananas = 35
Selling price of each banana = ₹5
To Find:
Total profit made.
Solution:
Step 1: Calculate the total cost price (CP).
Total CP = Quantity in kg Price per kg
Step 2: Calculate the total selling price (SP).
Total SP = Number of bananas Selling price per banana
Step 3: Calculate the profit.
Profit = Total SP - Total CP
Final Answer: Shyamala makes a profit of ₹85.
Q4Figure it Out (Chapter End)
A teacher placed textbooks that are 2.5 cm thick on a bookshelf. The teacher wanted to place 80 textbooks on the shelf. The bookshelf is 160 cm long. How many books could be placed on the shelf? Was there any space left? If yes, how much?
Solution
Given:
Thickness of one textbook = 2.5 cm
Length of the bookshelf = 160 cm
Number of books the teacher wanted to place = 80
To Find:
- How many books could be placed on the shelf.
- If there was any space left, and how much.
Solution:
Step 1: Calculate the number of books that can fit on the shelf.
Number of books = Total length of shelf Thickness of one book
So, 64 books could be placed on the shelf.
Step 2: Compare with the desired number of books.
The teacher wanted to place 80 books, but only 64 books fit. So, not all books could be placed.
Step 3: Calculate the space occupied by 64 books and the remaining space.
Space occupied = .
Space left = Total length of shelf - Space occupied
There was no space left after placing 64 books.
Final Answer: 64 books could be placed on the shelf. There was no space left on the shelf.
Q5Figure it Out (Chapter End)
Fill in the following blanks appropriately:
m m mm kg mm l
Solution
Conversion Factors:
Solutions:
Final Answer Table:
| m | m | mm |
|---|---|---|
| kg | mm | l |
Q6Figure it Out (Chapter End)
The following problem was set by Sridharacharya in his book, Patiganita. " is divided by , and is divided by . Tell the quotients separately." Can you try to solve it by converting the fractions into decimals?
Solution
Problem 1: divided by
Solution:
First, convert the mixed fractions to decimals.
Now, perform the division.
Final Answer for Problem 1: The quotient is 2.5.
Problem 2: divided by
Solution:
First, convert the mixed fractions to decimals.
Now, perform the division.
Performing long division:
Let's check with fractions:
Final Answer for Problem 2: The quotient is or approximately 17.214.
Q7Figure it Out (Chapter End)
Fill the boxes in at least 2 different ways:
(a)
□ × □
(b)
□ × □
Solution
(a) □ × □
Solution:
Here are two possible ways:
- (Other examples: , )
Final Answer:
Way 1:
Way 2:
(b) □ × □
Solution:
Here are two possible ways:
- (Other examples: , )
Final Answer:
Way 1:
Way 2:
Q8Figure it Out (Chapter End)
Find the following quotients given that :
(a)
(b)
(c)
(d)
(e)
(f)
Solution
Given:
(a)
Solution:
Final Answer: 21
(b)
Solution:
Final Answer: 21
(c)
Solution:
Final Answer: 2100
(d)
Solution:
Final Answer: 0.21
(e)
Solution:
Final Answer: 2100
(f)
Solution:
This is the same as part (b).
Final Answer: 21
Q9Figure it Out (Chapter End)
Find the missing cells if each cell represents :
b a 1517 151.7 15.17 1.517 15170 37 41 3.7 4.1 0.37 0.037 4100 370
Solution
Base Information:
From the table, we are given:
- All these are consistent with the base fact . For example, .
Solution:
We will fill the table by applying rules of decimal division based on the fact that the digits in the quotient will be 4 and 1.
-
Row 1 (b=37):
-
Row 2 (b=3.7):
-
Row 3 (b=0.37):
-
Row 4 (b=0.037):
-
Row 5 (b=370):
Final Answer (Completed Table):
| b a | 1517 | 151.7 | 15.17 | 1.517 | 15170 |
|---|---|---|---|---|---|
| 37 | 41 | 4.1 | 0.41 | 0.041 | 410 |
| 3.7 | 410 | 41 | 4.1 | 0.41 | 4100 |
| 0.37 | 4100 | 410 | 41 | 4.1 | 41000 |
| 0.037 | 41000 | 4100 | 410 | 41 | 410000 |
| 370 | 4.1 | 0.41 | 0.041 | 0.0041 | 41 |
Q10Figure it Out (Chapter End)
Using the digits 2, 4, 5, 8, and 0 fill the boxes to get the:
(a)
maximum product
(b)
minimum product
(c)
product greater than 150
(d)
product nearest to 100
(e) product nearest to 5
Solution
Note: The format of the multiplication is ambiguous. Based on typical problems and the targets given, we will assume the format is . The five digits 2, 4, 5, 8, and 0 must be used exactly once.
(a) maximum product
To maximize the product, we should place the largest digits in the highest place value positions. The numbers will be of the form AB.C and D.E.
Let's try combinations with 8 and 5 in the tens and units places.
Comparing these, the maximum product is from .
Answer (a):
(b) minimum product
To minimize the product, we should place the smallest digits in the highest place value positions.
Let's try combinations with 2 and 0 in the tens and units places.
Comparing these, the minimum product is from .
Answer (b):
(c) product greater than 150
From part (a), we found several combinations that work.
For example, , which is greater than 150.
Answer (c):
(d) product nearest to 100
Let's examine the products we calculated for the minimum value:
(Difference from 100 is 18.32)
(Difference from 100 is 1.6)
(Difference from 100 is 6.4)
The product nearest to 100 is 98.4.
Answer (d):
(e) product nearest to 5
With the assumed format , the minimum possible product is 93.6. It is not possible to get a product near 5.
Answer (e): Impossible with the assumed format. (If the format was , the answer would be )
Q11Figure it Out (Chapter End)
Sort the following expressions in increasing order:
(a)
(b)
(c)
(d)
(e) 245.05
(f) 7.9682
Solution
Given expressions:
Let , , and .
(a)
(b)
(c)
(d)
(e)
(f)
Reasoning:
- Multiplying a positive number () by a number less than 1 () gives a product smaller than . So, .
- Multiplying by a number greater than 1 () gives a product larger than . So, .
- Dividing by a number greater than 1 () gives a quotient smaller than . So, .
- Dividing by a number less than 1 () gives a quotient larger than . So, .
Estimation of values:
- (f)
- (c)
- (a)
- (e)
- (d) (approx 260)
- (b)
Ordering the estimated values:
Mapping back to the expressions:
(f) < (c) < (a) < (e) < (d) < (b)
Final Answer:
The expressions in increasing order are:
- (f) 7.9682
- (c)
- (a)
- (e) 245.05
- (d)
- (b)
Q1Figure it Out (Decimal Division - Part 1)
Find the quotient by converting the denominator into 1, 10, 100 or 1000 and verify the solution by the long division method (division by place value).
(a)
(b)
(c)
(d)
Solution
(a)
Method 1: Converting the denominator
To make the denominator 10, we multiply the numerator and denominator by 2.
Method 2: Long Division (Verification)
3.6
5 | 18.0 -15
3 0
-3 0
----
0
The quotient is 3.6. Both methods give the same result.
(b)
Method 1: Converting the denominator
To make the denominator 100, we multiply the numerator and denominator by 25.
Method 2: Long Division (Verification)
103.75
4 | 415.00 -4
01
- 0
---
15
-12
---
3 0
-2 8
----
20
-20
---
0
The quotient is 103.75. Both methods give the same result.
(c)
Method 1: Converting the denominator
To make the denominator 10, we multiply the numerator and denominator by 5.
Method 2: Long Division (Verification)
608.5
2 | 1217.0 -12
01
- 0
---
17
-16
---
1 0
-1 0
----
0
The quotient is 608.5. Both methods give the same result.
(d)
Method 1: Converting the denominator
To make the denominator 1000, we multiply the numerator and denominator by 125.
Method 2: Long Division (Verification)
603.375
8 | 4827.000 -48
02
- 0
---
27
-24
---
3 0
-2 4
----
60
-56
---
40
-40
---
0
The quotient is 603.375. Both methods give the same result.
Q2Figure it Out (Decimal Division - Part 1)
Choose the correct answer: (a)
(i)
38.15
(ii)
380.15
(iii)
381.5
(iv)
381.05
(b)
(i)
4458.75
(ii)
44.5875
(iii)
445.875
(iv)
4458.75
Solution
(a)
Solution:
We perform the long division of 1526 by 4.
381.5
4 | 1526.0 -12
32
-32
06
-4
--
2 0
-2 0
----
0
The quotient is 381.5.
Answer: (iii) 381.5
(b)
Solution:
We perform the long division of 3567 by 8.
445.875
8 | 3567.000 -32
36
-32
47
-40
---
7 0
-6 4
----
60
-56
---
40
-40
---
0
The quotient is 445.875.
Answer: (iii) 445.875
Q3Figure it Out (Decimal Division - Part 1)
What is the quotient?
(a)
(b)
(c)
(d)
Solution
(a)
Solution:
Final Answer: 33
(b)
Solution:
Since , and the dividend 13.2 has one decimal place, the quotient will also have one decimal place.
Final Answer: 3.3
(c)
Solution:
Since , and the dividend 1.32 has two decimal places, the quotient will also have two decimal places.
Final Answer: 0.33
(d)
Solution:
Since , and the dividend 0.132 has three decimal places, the quotient will also have three decimal places. We need to add a leading zero.
Final Answer: 0.033
Q4Figure it Out (Decimal Division - Part 1)
What is the quotient?
(a)
(b)
(c)
(d)
(e)
Solution
(a)
Solution:
Final Answer: 15.75
(b)
Solution:
Since , and the dividend 12.6 has one decimal place, we shift the decimal in the quotient 15.75 one place to the left.
Final Answer: 1.575
(c)
Solution:
Since , and the dividend 1.26 has two decimal places, we shift the decimal in the quotient 15.75 two places to the left.
Final Answer: 0.1575
(d)
Solution:
Since , and the dividend 0.126 has three decimal places, we shift the decimal in the quotient 15.75 three places to the left.
Final Answer: 0.01575
(e)
Solution:
Since , and the dividend 0.0126 has four decimal places, we shift the decimal in the quotient 15.75 four places to the left.
Final Answer: 0.001575
Q1Figure it Out (Decimal Division - Part 2)
Express the following fractions in decimal form:
(a)
(b)
(c)
(d)
Solution
(a)
Solution:
To convert to a decimal, make the denominator a power of 10.
Final Answer: 0.4
(b)
Solution:
Alternatively, .
Final Answer: 3.25
(c)
Solution:
Final Answer: 0.08
(d)
Solution:
Alternatively, .
Final Answer: 0.625
Q2Figure it Out (Decimal Division - Part 2)
Find the quotients:
(a)
(b)
Solution
(a)
Solution:
To make the divisor a whole number, multiply both dividend and divisor by 10.
Now, we perform long division:
20.7166...
12 | 248.6000
-24
---
08
- 0
---
8 6
-8 4
----
20
-12
---
80
-72
---
80
-72
---
8
The quotient is approximately 20.717.
Final Answer: 20.716... (approximately 20.72)
(b)
Solution:
To make the divisor a whole number, multiply both dividend and divisor by 100.
Now, we perform long division:
3.768...
152 | 572.800
-456
-----
116 8
-106 4
------
104 0
-91 2
-----
128 0
-121 6
------
64
The quotient is approximately 3.77.
Final Answer: 3.768... (approximately 3.77)
Q3Figure it Out (Decimal Division - Part 2)
Evaluate the following using the information .
(a)
(b)
(c)
(d)
Solution
Given:
(a)
Solution:
The numbers are the same as the given information, but with decimal points. Total decimal places = 1 + 1 = 2.
So, .
Final Answer: 18.72
(b)
Solution:
From the given information, we know . Let's adjust for decimals.
Final Answer: 156
(c)
Solution:
From the given information, we know . Let's adjust for decimals.
Final Answer: 1.2
(d)
Solution:
Total decimal places = 3 + 2 = 5.
So, .
Final Answer: 0.01872
Q4Figure it Out (Decimal Division - Part 2)
Evaluate the following:
(a)
(b)
(c)
(d)
(e)
(f)
Solution
(a)
Solution:
Final Answer: 1000
(b)
Solution:
Final Answer: 0.1
(c)
Solution:
Final Answer: 10
(d)
Solution:
Dividing by a number is the same as multiplying by its reciprocal. The reciprocal of 10 is or 0.1.
Final Answer: 0.1 or
(e)
Solution:
The reciprocal of 0.10 (or 0.1) is .
Final Answer: 10
(f)
Solution:
The reciprocal of 0.01 is .
Final Answer: 100
Q5Figure it Out (Decimal Division - Part 2)
Find the quotient:
(a)
(b)
(c)
Is the quotient obtained in the same as the quotient obtained in ?
Solution
(a)
Solution:
Final Answer: 1.64
(b)
Solution:
Final Answer: 16.4
(c)
Solution:
Final Answer: 164
Comparison:
Is the quotient obtained in the same as the quotient obtained in ?
Let's calculate .
From part (b), we found that .
Final Answer: Yes, the quotient is the same in both cases. This is because in both divisions, the dividend and divisor were multiplied by the same power of 10 to get an equivalent fraction: and .
Q6Figure it Out (Decimal Division - Part 2)
A 4 m long wooden block has to be cut into 5 pieces of equal length. What is the length of each piece?
Solution
Given:
Total length of the wooden block = 4 m
Number of equal pieces = 5
To Find:
Length of each piece.
Solution:
Length of each piece = Total length Number of pieces
To express this as a decimal:
Final Answer: The length of each piece is 0.8 m.
Q7Figure it Out (Decimal Division - Part 2)
If the perimeter of a regular polygon with 12 sides is 208.8 cm, what is the length of its side?
Solution
Given:
Perimeter of a regular polygon = 208.8 cm
Number of sides = 12
To Find:
Length of one side of the polygon.
Solution:
In a regular polygon, all sides are of equal length.
Length of one side = Perimeter Number of sides
Final Answer: The length of its side is 17.4 cm.
Q8Figure it Out (Decimal Division - Part 2)
3 litres of watermelon juice is shared among 8 friends equally. How much watermelon juice will each get? Express the quantity of juice in millilitres.
Solution
Given:
Total quantity of juice = 3 litres
Number of friends = 8
To Find:
Quantity of juice each friend will get, in millilitres.
Solution:
First, find the share per friend in litres.
Share per friend = Total juice Number of friends
Now, convert this quantity to millilitres. We know that .
Quantity in ml =
Final Answer: Each friend will get 375 ml of watermelon juice.
Q9Figure it Out (Decimal Division - Part 2)
A car covers 234.45 km using 12.6 litres of petrol. What is the distance travelled per litre?
Solution
Given:
Total distance covered = 234.45 km
Total petrol used = 12.6 litres
To Find:
Distance travelled per litre (mileage).
Solution:
Distance per litre = Total distance Total petrol
Performing the division:
Rounding to two decimal places, the distance is 18.61 km per litre.
Calculation:
18.607
126|2344.500 -126
1084
-1008
76 5
-75 6
-----
90
-0
--
900
-882
----
18
Final Answer: The distance travelled per litre is approximately 18.61 km.
Q10Figure it Out (Decimal Division - Part 2)
13.5 kg of flour (aata) was distributed equally among 15 students. How much flour did each student receive?
Solution
Given:
Total amount of flour = 13.5 kg
Number of students = 15
To Find:
Amount of flour each student received.
Solution:
Flour per student = Total flour Number of students
Final Answer: Each student received 0.9 kg of flour.
Q1Figure it Out (Decimal Multiplication)
Recall that a tenth is 0.1 , a hundredth is 0.01 , and so on. Find the following products in tenths, hundredths and so on:
(a)
tenths tenths
(b)
(c)
hundredths
Solution
(a) tenths tenths
Solution:
4 tenths can be written as .
2.4 is equal to 24 tenths, since .
Final Answer: 24 tenths or 2.4
(b)
Solution:
is 3 tenths.
21 tenths is equal to .
Final Answer: 21 tenths or 2.1
(c) hundredths
Solution:
5 hundredths can be written as .
0.45 is equal to 45 hundredths, since .
Final Answer: 45 hundredths or 0.45
Q2Figure it Out (Decimal Multiplication)
Find the products:
(a)
(b)
(c)
Solution
(a)
Solution:
First, multiply 2734 by 6:
In 27.34, there are 2 digits after the decimal point. So, in the product, we place the decimal point 2 places from the right.
Final Answer: 164.04
(b)
Solution:
First, multiply 423 by 37:
In 4.23, there are 2 digits after the decimal point. In 3.7, there is 1 digit after the decimal point. Total number of decimal places is .
So, in the product, we place the decimal point 3 places from the right.
Final Answer: 15.651
(c)
Solution:
First, multiply 432 by 23:
In 0.432, there are 3 digits after the decimal point. In 0.23, there are 2 digits after the decimal point. Total number of decimal places is .
So, in the product, we place the decimal point 5 places from the right. We need to add a zero in front.
Final Answer: 0.09936
Q3Figure it Out (Decimal Multiplication)
Thejus needs 1.65 m of cloth for a shirt. How many metres of cloth are needed for 3 shirts?
Solution
Given:
Cloth needed for 1 shirt = 1.65 m
Number of shirts = 3
To Find:
Total cloth needed for 3 shirts.
Solution:
Total cloth needed = Cloth for 1 shirt Number of shirts
Final Answer: 4.95 metres of cloth are needed for 3 shirts.
Q4Figure it Out (Decimal Multiplication)
Meenu bought 4 notebooks and 3 erasers. The cost of each book was ₹15.50 and each eraser was ₹2.75. How much did she spend in all?
Solution
Given:
Number of notebooks = 4
Cost of each notebook = ₹15.50
Number of erasers = 3
Cost of each eraser = ₹2.75
To Find:
Total amount spent by Meenu.
Solution:
Cost of 4 notebooks =
Cost of 3 erasers =
Total amount spent = Cost of notebooks + Cost of erasers
Final Answer: Meenu spent ₹70.25 in all.
Q5Figure it Out (Decimal Multiplication)
The thickness of a rupee coin is 1.45 mm . What is the total height of the cylinder formed by placing 36 rupee coins one over the other? Write the answer in centimeters.
Solution
Given:
Thickness of one coin = 1.45 mm
Number of coins = 36
To Find:
Total height of 36 coins in centimeters.
Solution:
Total height in mm = Thickness of one coin Number of coins
We know that , so .
To convert mm to cm, we divide by 10.
Total height in cm =
Final Answer: The total height of the cylinder is 5.22 cm.
Q6Figure it Out (Decimal Multiplication)
The price of 1 kg of oranges is ₹56.50. What is the price of 2.250 kg of oranges? Can we write 56.50 as 56.5 and 2.250 as 2.25 and multiply? Will we get the same product? Why?
Solution
Given:
Price of 1 kg of oranges = ₹56.50
Weight of oranges to buy = 2.250 kg
To Find:
Price of 2.250 kg of oranges and justification for simplifying decimals.
Solution:
Price of 2.250 kg of oranges = Price per kg Weight
So, the price is ₹127.125.
Now, let's consider the second part of the question.
Yes, we can write 56.50 as 56.5 and 2.250 as 2.25.
Let's multiply these simplified numbers:
Yes, we get the same product.
Reason:
Trailing zeros after the decimal point do not change the value of the number. For example:
Since the values of the numbers are the same, their product will also be the same.
Final Answer: The price of 2.250 kg of oranges is ₹127.125. Yes, we can write 56.50 as 56.5 and 2.250 as 2.25 and multiply, and we will get the same product because trailing zeros after the decimal point do not change the value of the number.
Q7Figure it Out (Decimal Multiplication)
Dwarakanath purchases notebooks at a wholesale price of ₹23.6 per piece and sells each notebook at ₹30/-. How much profit does he make if he sells 50 books in a week?
Solution
Given:
Wholesale price (Cost Price) of one notebook = ₹23.6
Selling price of one notebook = ₹30.00
Number of books sold = 50
To Find:
Total profit from selling 50 books.
Solution:
First, find the profit on one notebook.
Profit per notebook = Selling Price - Cost Price
Now, calculate the total profit for 50 books.
Total Profit = Profit per notebook Number of books sold
Final Answer: Dwarakanath makes a profit of ₹320 if he sells 50 books in a week.
Q8Figure it Out (Decimal Multiplication)
Given that , find the products:
(a)
(b)
(c)
(d)
(e)
(f)
In which of the cases above is the product less than 1?
Solution
Given:
Solution:
We use the given product 216 and place the decimal point based on the total number of decimal places in the factors.
(a)
Total decimal places = 1. So,
(b)
Total decimal places = 2. So,
(c)
Total decimal places = 1 + 1 = 2. So,
(d)
Total decimal places = 2 + 2 = 4. So,
(e)
Total decimal places = 3 + 3 = 6. So,
(f)
Total decimal places = 1. So,
Product less than 1:
A product is less than 1 if both factors are between 0 and 1, or if the product itself evaluates to a value less than 1.
Looking at the results:
(a) 21.6 > 1
(b) 2.16 > 1
(c) 2.16 > 1
(d) 0.0216 < 1
(e) 0.000216 < 1
(f) 21.6 > 1
Final Answer:
(a) 21.6
(b) 2.16
(c) 2.16
(d) 0.0216
(e) 0.000216
(f) 21.6
The products in cases (d) and (e) are less than 1.
Q9Figure it Out (Decimal Multiplication)
In which of the following multiplications is the product less than 1? Can you find the answer without actually doing the multiplications?
(a)
(b)
(c)
(d)
Solution
Rule: The product of two numbers that are both between 0 and 1 is always less than both of the original numbers (and therefore less than 1).
(a)
One number is greater than 1 (7) and one is less than 1 (0.6). The product will be between 0.6 and 7. Specifically, , which is greater than 1.
(b)
Both numbers (0.7 and 0.6) are less than 1. Therefore, their product will be less than 1. ()
(c)
One number is less than 1 (0.7) and one is greater than 1 (6). The product will be between 0.7 and 6. Specifically, , which is greater than 1.
(d)
Both numbers (0.07 and 0.06) are less than 1. Therefore, their product will be less than 1. ()
Final Answer: The products in (b) and (d) are less than 1.
Q10Figure it Out (Decimal Multiplication)
Multiplying the following numbers by 10, 100 and 1000 to complete the table.
5.7 23.02 0.92 0.306 24.67
Solution
Rule: To multiply a decimal by 10, 100, 1000, etc., move the decimal point to the right by the same number of places as there are zeros in the multiplier.
Completed Table:
| 5.7 | 57 | 570 | 5700 |
| 23.02 | 230.2 | 2302 | 23020 |
| 0.92 | 9.2 | 92 | 920 |
| 0.306 | 3.06 | 30.6 | 306 |
| 24.67 | 246.7 | 2467 | 24670 |