Arithmetic
'Percent' means 'for every hundred'.
Symbol for percentage is %.
a) Percentage to decimals:
To convert a percentage to a decimal, divide the number by 100.
e.g., 68% = \[\frac{68}{100}\] = 0.68
b) Decimal to percentage:
To convert a decimal to a percentage, multiply the number by 100%.
e.g., \[0.59 = 0.59\times 100% = 59%\]
c) Percentage to fraction:
To convert a percentage to a fraction, write the number with denominator 100 and reduce the fraction to its lowest terms.
e.g., 45% =\[\frac{45}{100}=\frac{9}{20}\]
d) Fraction to percentage:
To convert a fraction to a percentage, multiply one fraction with 100%
e.g., \[\frac{9}{20}=\frac{9}{20}\times 100%=45%\]
e) Finding the percent of a quantity:
To find the percent of a quantity, multiply them and simplify.
e.g., 30% Rs 100
\[\frac{30}{100}\times \text{Rs}100=\text{Rs30}\]
\[\text{Average = }\frac{\text{The sum of quantities}}{\text{The number of quantities}}\]
(a) The comparison of two quantities of the same kind by division gives their ratio.
(b) The two quantities compared are written with a : (colon) between them.
e.g., a; b read as 'a is to b'.
(c) Ratio of two numbers can be thought of as a fraction and all the rules for operations with fractions can be used.
(d) Double, triple, four times, etc., can be expressed in ratio as 2:1, 3:1, 4:1, etc.
(e) A ratio can be expressed as a fraction.
e.g., 2: 5 is the same as \[\frac{2}{5}\].
(f) In a ratio a: b, the first term 'a' is called the antecedent and the second term 'b' is called the consequent. The order of terms of a ratio is important i.e., 1:4 is not the same ratio as 4:1.
(g) To find the ratio of two like quantities, they should be changed into the same unit of measurement.
(h) While writing a ratio, co-prime numbers are generally used, that is, the ratio is often expressed in the lowest terms by cancelling the common factors from both the numbers.
(i) A ratio does not have any unit of measurement.
- Speed, Distance and Time:
\[\text{Speed =}\frac{\text{Distance}}{\text{Time}}\]
\[\text{Average }=\text{ }\frac{\text{Total distance covered}}{\text{Total time taken}}\]
\[\operatorname{Distance} = Speed\times Time\]
\[\text{Time }=\text{ }\frac{\text{Distance }}{\text{Speed}}\]
|\[=\frac{\text{PTR}}{100}\], where |= Interest, P = Principal, T = Time, R = Rate per annum
Amount (A) = P + | \[\Rightarrow \]| = A -P and also P = A -|
(i) The price of an article is called its cost price denoted as C.R
(ii) The price at which an article is sold is called its selling price denoted as S.R
(iii) If the selling price is greater than the cost price, there is a gain/profit, which is equal to the difference of selling price and cost price.
∴ If S.P. > C.R,
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(b) Mass:
(c) Capacity:
(d) Time:
To express length, mass or capacity using bigger unit, we use decimals.
Factors divide the number exactly (i.e., without leaving a remainder.) So, factors are also called divisors.
1 is a factor of every number, and every number is a factor of itself.
1 is the smallest factor of a number and the number itself is its greatest factor.
The factor of a number is less than or equal to the number.
Every number (except 1) has at least 2 factors - 1 and the number itself.
The factors or divisors that are common to two or more numbers are called their common factors.
∴ Common factors of 4 and 6 are 1, 2.
3×100 = 300 and 75×4=300
is equivalent to \[\frac{75}{100}\]
e.g. (b) \[\frac{1}{5},\frac{2}{3}\]
1×3=3 and 2×5=10
3 ≠ 10
∴ \[\frac{1}{5}\] is not equivalent to\[\frac{2}{3}\].
A line \[\overleftrightarrow{\text{AB}}\] is read as line AB'.
\[\overleftrightarrow{\text{AB}}\]=\[\overleftrightarrow{\text{BA}}\]
Line segment
A line segment PQ is written as PQ.
\[\overline{\text{PQ }}\text{= }\overline{\text{QP}}\]
A ray OP is written as 0?. A ray is denoted by writing the initial point first.
So, \[\overrightarrow{\text{OP}}\ne \overrightarrow{\text{PO}\text{.}}\]
\[\overrightarrow{\text{OA}}\] and \[\overrightarrow{\text{OB}}\] form an angle AOB. 0 is the vertex of \[\angle \text{AOB}\]and \[\overrightarrow{\text{OA}}\]and \[\overrightarrow{\text{OB}}\] are its arms or sides.
Z AOB is the same as Z BOA. Only vertex can also be used to denote an angle. Thus ZA means the angle whose vertex is A.
Types of Angles:
(a) An angle whose measure is between 0° and 90° is called an acute angle.
(b) An angle whose measure is 90° is called a right angle.
(c) An angle whose measure is between 90° and 180° is called obtuse angle.
In the given figure, B, F, and E are collinear, while A, C and D are non-collinear points.
The centre of a circle is usually denoted as 0.
It is a solid figure with 6 rectangular surfaces.
The net of a cube has 6 squares.
The net of a cuboid has 6 rectangles.
Human body has the following systems performing certain functions:
Rib Cage
Ribs make a cage of bones around the chest which is called rib cage. It protects our internal organs. Generally adults have 12 pairs of ribs. There is a long bone at the centre of the chest, which holds the ribs in place, called sternum. Ribs are attached to the backbone. Last two ribs are not attached to the sternum and are known as floating ribs. These floating ribs are attached to the backbone.
Backbone
It is made up of 33-ring shaped small bones called vertebrae. Backbone is also called as vertebral column. It is a hollow tube through which spinal cord runs.
Limbs
All human beings have two pairs of limbs: the forelimbs (arms) and hindlimbs (legs). Both forelimb and hindlimb is made up of 30 bones each. Thigh bone or femur is the longest bone in the body.
Note: The smallest bone of the body called stapes is present in the ear.
Functions of the Skeletal System
The skeletal system has the following functions:
