Math Tutoring
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Course Summary:👉
Subjects:
- Fraction
- Decimals
- Integers
- Exponents
- Inequalities
- Percent
- Square Roots
- Ratios and Proportion
- Ratios
- Proportions
- Distance, rates, and time
- Similar figures
Up next for you:
Course challenge👭
Test your knowledge of the skills in this course.
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what you need to review.
https://www.educatorsharmin.com/search/label/Math%20Class?&max-results=5
https://www.youtube.com/watch?v=hT8OD7SGvRM
Contents:
1. Large Numbers
2. Operations involving Large Numbers
3. Rounding Off Numbers
4. Multiples and Factors
5. Operations on Fractions
6. Operations on Decimal Numbers
7. Percentage
8. Average
9. Unitary Method and Ratio & Proportion
10. Bills
13. Speed, Distance & Time
14. Geometry
17. Temperature
18. Bar Graphs & Pie Charts
19. Timetables
20. Algebra Readiness
Chapter -1
Large Numbers
- The place value of a digit is the value of the digit because of its place in the number.
- Starting from the right side of the number, according to the Indian system of numeration, the first period is the ones period consisting of three place values- ones, tens and hundreds.
- The other periods to the left of the ones period have two place value each.
International Place-Value
A different form of place-value chart known as the International Place-value Chart is used in most other countries of the world.
Billions Millions Thousands Ones
B HM TM M HTh TTh Th H T O
Rounding Number:
* When we round a number to the nearest ten, we round it to the multiple of ten nearest to it.
* A number which is midway is always rounded up.
* When we round a number to the nearest hundred, we round it to the multiple of hundred nearest to the number.
* A number which is midway is always rounded up.
Roman Numerals:
There is no zero in the Roman system. An important difference between the decimal system and the Roman system is that the Roman system does not use place value.
Rules:
1. Numerals I, X, C and M can be repeated to represent a number. Repetition means addition. I, X and C can not be repeated more than 3 times. Symbols V, L, D are not repeated.
Thus, III = 3
XXX = 30
CCC = 300
2. A smaller number written to the right of the numeral of greater value is always added to the greater numeral.
VI = 5 + 1 = 6
XV = 10 + 5 = 15
LX = 50 + 10 = 60
CX = 100 + 10 = 110
3. A smaller number written to the left of a number of greater value is always subtracted from the greater numeral.
IV = 5 - 1 = 4
XL = 50 - 10 = 40
XC = 100 - 10 = 90
4. When a smaller number is placed between two numerals of greater value, it is always subtracted from the greater numeral immediately following it.
XIV = 10 + (5 - 1 )
= 14
XXIX = 10 + 10 + ( 10 -1 )
= 29
CXXIV = 100 + 10 + 10 + (5 - 1 )
= 124
Chapter-2
Operations involving Large Numbers
Word Problems:
1. China has an area of 95,72,900 sq. km (approx.) and India is spread over 32,87,263 sq. km (approx). What is the area of the two countries taken together?
Solution:
C TL L TTh Th H T O
China 9 5 7 2 9 0 0 sq. km
India + 3 2 8 7 2 6 3 sq. km
___________________________________________________________________ 1 2 8 6 0 1 6 3 sq. km
The area of India and China has taken together is 1,28,60,163 sq. km (approx).
2. When the planets Mercury and Venus are in line with the Sun, Mercury is 5,80,00,000 km from the Sun whereas Venus is 10,80,00,000 km from the Sun. What is the distance between Mercury and Venus?
Solution:
Distance of Venus from Sun 1080,00,000 km
Distance of Mercury from Sun -5,80,00,000 km
________________________________________
Distance between Mercury and Venus 50000000 km
Thus, the distance between Mercury and Venus is 50000000 km
3. The population of the USA is 30,11,39,947 whereas the population of Mexico is 10,87,00891 and that of Canada is 3,33,90,141. By how much is the population of the USA greater than the population of Canada and Mexico taken together?
Population of Mexico 10,87,00891
Population of Canada + 3,33,90,141
____________________________________________
The population of Mexico & Canada 142091032
Population of the USA 301139947
Population of Mexico & Canada - 142091032
________________________________________________________
Population of the USA is greater by 159048915
Thus, the population of the USA is greater by 15,90,48,915.
Rules of using Brackets:
Rule 1: Calculate from left to right.
Rule -2: Do division first then multiplication, and then do addition and subtraction.
Rule -3: Calculate inside the brackets first. Open the brackets from the first bracket ( ) to the second bracket { }, and to the third bracket [ ].
Complete order of Operations:
We now understand that we have to perform operations within brackets and multiply numbers with 'of' in between before we do the four operations. We can sum up this order by using a popular acronym.
1st operation B Brackets [ Simplify all operations within ( ) followed by { } ]
1st operation B Brackets
2nd operation O Of [ Multiply the 2 numbers containing 'of' ]
3rd operation D Division
4th operation M Multiplication
5th operation A Addition
6th operation S Subtraction
Simplify:
26 + {(6 ➗ 2 - 1) + 4 } ➗3 of 2 - 3 x 9
= 26 + {(3 - 1) + 4 } ➗3 of 2 - 3 x 9
= 26 + {2 + 4 } ➗3 of 2 - 3 x 9
= 26 + 6 ➗ 6 - 3 x 9
= 26 + 1 - 3 x 9
= 0 Ans.
Chapter- 3
Rounding Off Numbers
If the digit at the lower place value is less than 5, we round off the given number by rounding down. i.e. by keeping the digit at the desired place same and writing '0' in all the lower place values.
If the digit at the lower place value is equal to or greater than 5, we round off the given number by rounding up. i.e. by adding 1 to the digit at the desired place value and writing '0' in all the lower place values.
Example:
371 = 370
7645 = 7650
* To round off a number to the nearest desired place value, we should consider the digit at the next lower place value.
Example:
52789604 = 5,000,000
Chapter- 4
Multiples and Factors
The products of a number and counting numbers are known as the multiples of that number.
* Every number is a multiple of itself.
* Every number is a multiple of 1.
* There is no limit to the number of multiples of a number.
* Every multiple of a number is greater than or equal to the number.
The numbers that are multiplied to get a product are called the factors of the product.
* Every number is a factor of itself. It is also the greatest factor of that number.
* 1 is a factor of every number. It is also the smallest factor of a number.
* The factor of a number is less than or equal to the number.
A prime number is a number which has no factors other than itself and unity.
The first five prime numbers are 2, 3, 5, 7, 11.
A prime factor is a factor which is a prime number.
The prime factors of 24 are 2, 2, 2, and 3.
Expressed in its prime factors ,
24 = 2 x 2 x 2 x 3
Ex. 7.4
A number that is not a prime number is a composite number.
A composite number has at least one factor other than 1 and the number itself.
Two numbers that have only 1 as a common factor are known as coprime numbers.
Example:
Are 81 and 256 coprime numbers?
First of all, express both the numbers as the product of prime numbers and 1.
81 = 3 x 3 x 3 x 3 x 1
256 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
We observe that both the numbers are composite numbers.
But the only common factor between them is 1.
Thus, 81 and 256 are coprime numbers.
Highest Common Factor:
The Highest Common Factor (H.C.F) of two or more numbers is the greatest number which is a factor of each of them.
Ex. Find the H.C.F and L.C.M. of 63, 441, 3969.
4. List all the common factors of the following numbers. also write the highest common factors (HCF) for each pair.
a) 9, 15
Solution: 9 = 1 x 9 = 3 x 3
The factors of 9 : 1, 3, 9
15 = 1 x 15
= 3 x 5
The factors of 15 : 1 ,3, 5, 15
The common factors of 9 and 15 are 1 and 3.
The highest common factors (HCF) for 9 and 15 is 3. Ans.
Lowest Common Multiple:
The Lowest Common Multiple (L.C.M.) of two or more numbers is the least number which is a multiple of each of them.
The numbers by which a larger number can be divided are the factors of that number.
When a number is divisible by another number, the first number is called the multiple of the other number.
2. List 3 common multiples for the following pairs of numbers, ordering them from the smallest. also write the least common multiples (LCM).
a) 3, 4
Solution: Multiples of 3 : 3, 4, 6, 9, 12, 15, 18, 24, 27, 30, 33, 36,............
Multiples of 4 : 4, 8, 12, 16, 20, 24, 28, 32, 36,...........
3 common multiples of 3 and 4 are 12, 24, 36
Ordering from smallest to highest 12, 24, 36
The smallest common multiple is 12
The LCM of 3 and 4 = 12 Ans.
Chapter- 5
Operations on Fractions
Addition:
Convert the fractions into equivalent fractions with the same denominator by multiplying both the numerator and denominator by the same number.
Add the numerators of the fractions with the same denominator.
Simplify:
5/6 + 1/2
convert the fractions into equivalent fractions with the same denominator.
5/6 + 1/2 = 5/6 + 3/6
Add the numerators of the fractions with the same denominator.
5/6 + 3/6 = 5+3/6 = 8/6 = 4/3
Subtraction:
Convert the fractions into equivalent fractions with the same denominator by multiplying both the numerator and denominator by the same number.
Subtract the numerator of the second fraction from the first denominator.
Example:
Subtract 1/5 from 3/10.
Convert the fractions into equivalent fractions with the same denominator.
3/10 - 1/5 = 3/10 - 1x2/5x 2 = 3/10 - 2/10
Subtract the numerators of the fractions with the same denominator.
3/10 - 2/10 = 3-2 / 10 = 1/10
Chapter- 6
Operations On Decimal Numbers
* A decimal has two parts - whole part
decimal part
* Like decimal numbers have an equal number of digits to the right of the decimal point.
* To add or subtract decimals, convert the decimals to like decimals, convert the decimals to like decimals. The decimal numbers are arranged vertically with the decimal point in one column, and then each column is added or subtracted.
Example:
Multiply: 0.3 x 0.2
Multiplicand 0.3
Multiplier 0.2
__________________________
0.6
000
____________________________________
Product 0.06
To convert a decimal number to a common fraction, follow the given steps.
1. Count the digits to the right of the decimal point.
2. Ignore the decimal point and write all the digits as the numerator of the fraction.
3. Write as many zeros after 1 in the denominator as the number of digits after the decimal point.
4. Reduce simple to the simplest form.
Express 3.649 as a fraction.
Number of digits to the
right of the decimal point = 3
Numerator = 3649
Denominator = 1000
Fraction = 3649/1000
Chapter -7
Percentage
Percentages are useful in conveying information in everyday life. The symbol, % is used to represent 'percent'.
Thus 50 % is read as '50 percent.
Suppose the sales of a company last year was $ 100 million and the sales of the company this year is $200 million. The sales of the company this year is 200/100 = 200% of its sales last year.
Expressing a Percentage as a Fraction:
a) Express each of the following percentages as a fraction.
i) 88%
= 88/100
= 22/25
ii) 128 %
= 128 /100
= 32/25
iii) 0.5%
= 0.5/100
= 1/200
b) Express each of the following fractions as a percentage.
i) 1/2
= 1/2 X 100%
= 50%
ii) 9/8
= 9/8 X 100%
= 900/8 %
= 112.5 %
a) Express each of the following percentages as a fraction.
i) 45 %
= 45/100
= 9/20
b) Express each of the following fractions as a percentage.
i) 17/20
= 17 / 20 X100
= 85 %
Q.2. Express each of the following percentages as a decimal.
i) 85%
= 85 / 100
= 0.85
Express each of the following decimals as a percentage.
i) 0.16
= 0.16 X 100%
= 16%
Percentage Word Problem
Q.1. A class has 40 students. Given that 75% of them passed a Mathematics test. Calculate the number of students who failed the test.
Solution:
Percentage of students who failed the test = 100% - 75 %
= 25%
Number of students who failed the test = 25% x 40
= 0.25 x 40
= 10 Ans.
iv) Converting Percentage to Decimal numbers:
Represent 60% as a decimal number 60 % = 60 out of 100
= 60/100
= 6/10
= 0.6
Q.2. A grocer buys a sack of rice for $ 840 and was able to sell it for $820. Find the grocer’s gain or loss.
Solution:
Selling Price = $ 820 Cost Price
= $840
As C.P. is more than S.P. Loss
= $840 - $820
= $20 Ans
Average
In our daily conversations, we often use the word ‘average’. Without really understanding the meaning of it. The meaning is really simple.
An average is a single number that gives an indication of most numbers or quantities in a group.
Consider the following situation.
Following are the marks obtained by a group of students of a particular class in a mathematics exam.
The sum of the marks obtained by the five students is 400. If we divide the sum of the marks by 5, we get the average marks obtained by the five students.
Average marks obtained = 400 / 5 = 80
This does not mean that all the students scored exactly 80 marks. Some of the students have scored more than 80 marks while some of them scored less than 80 marks, but by and large, most students scored around 80. The average marks tell us how many marks each student would have scored if all students scored the same marks.
The average lies between the greatest and the least quantity.
The average of a given number of quantities need not be a value in the given set of quantities.
Average cannot be calculated from two or more different types of quantities.
An average is a single number that gives an indication of most numbers or quantities in a group.
The average of two or more quantities is the sum of all the quantities divided by the number of quantities.
Average = Sum of the quantities / Number of quantities
Q. The weight of 6 books is 924 grams. Find the average weight of these books.
Solution:
The weight of 6 books is 924 gram
The average weight of books = ( 924 / 6 ) gram
= 154 gram Ans.
Chapter- 9
Unitary Method & Ratio & Proportion
A ratio is used to compare two or more quantities of the same kind which are measured in the same unit.
The ratio of a is to b, where a and b represent two quantities of the same kind, and b not = 0, is written as a:b.
Word Problem:
There are 17 boys and 19 girls in a class. Find the ratio of
i) the number of boys to the number of girls,
ii) the number of girls to the number of boys.
Solution:
Ratio of the number of boys to the number of girls = 17 : 19
Ratio of the number of girls to the number of boys = 19 : 17
1:2, 2:4 and 6 : 12 are known as equivalent ratios.
Q.2. If x : y = 2:3, find an equation connecting x and y.
x/y = 2/3
3x = 2y
x = 2/3 y
3. A farmer owns 50 head of cattle and 225 sheep. Find the ratio of the number of cattle to the number of sheep. Find also the ratio of the number of sheep to the total number of cattle and sheep.
Solution:
Cattle = 50
Sheep = 225
Cattle : Sheep = 50 : 225
= 50/225
= 2/9 Ans.
Total number of Cattle and Sheep = 50 + 225
= 275
Sheep : total Cattle & Sheep
= 225 : 275
= 225/275
= 9/11 Ans.
4. If each side of a square is enlarged three times, by how many times is its area increased?
Solution:
Suppose each side of the square is = a
It's area = a2
If each side is enlarged 3 times, then each side will be = 3a
So the area of the new square
= (3a)2
= 9a2
So, the ratio = 9a2: a2
= 9a2/a2
= 9 /1
= 9:1 Ans.
5. Express the ratio 2.5 : 8.0 in the form 1:n.
Solution: 2.5 : 8.0
= 2.5/8.0
= 25/80
= 5/16
= 1 : 3.2 Ans.
6. A map is drawn on a scale of 2 cm to 1 km. Express its representative fraction in the form 1:n.
Solution:
The ratio is
2 cm : 1km
= 2cm / 1km
= 2cm / 100000 cm
= 1/ 50000
= 1 : 50000 Ans.
Proportion:
In a proportion, the product of the extremes is equal to the product of the means.
Example: Is the ratio 5:4 in proportion to 25 : 20?
First ratio = 5:4
Second ratio = 25 : 20
The two may be written as a proportion as 5:4 :: 25 : 20
Product of extremes = 5 x 20 = 100
Product of means = 4 x 25 = 100
Since the product of extremes is equal to the product of means, the ratios 5:4 and 25 : 20 are in proportion.
Chapter- 10
Bills
A bill provides information about items or services purchased for which payment is to be made.
* A bill lets the purchaser check the supplier's calculations and varify if he is getting the quantities required at the prices agreed upon.
* In case of any dispute or when seeking refunds for damaged goods, etc. a bill is an important document, being the proof of purchase.
Chapter- 13
Speed, Distance & Time
The distance travelled in unit time by an object is called its speed.
Average speed is defined as distance travelled per unit time.
For example, if you cycle 2.4 km in 10 minutes, then your average speed would be
2400 m/ 10 x 60 s
= 24/6 m/s
= 4 m/s
1. The distance between Mumbai and Pune is 180 km. A bus takes about 4 hours to cover the distance. What is the average speed of the bus?
Solution:
Average speed = Distance / time
= 180/4 km/h
= 45 km/h
The most commonly used units for speed are km/h and m/s.
Speed = Distance/Time
Distance = Speed x Time
Time = Distance covered / Speed
Average speed is defined as distance travelled per unit time. It is calculated when the distance travelled per hour is not uniform due to stoppage, slowing down etc.
2. A train travels for 3 hours at a speed of 121 km/h. How much distance does it cover in that time?
Applying the unitary method, we get:
In 1 hour, the train travels 121 km
In 3 hours, the train travels 121 x 3 = 363 km
Thus, we observe that
Distance = Speed x Time
