Current Affairs 8th Class

Category : 8th Class

 

Algebra

           

Learning Objectives

 

• Liner equation in one variable
• Algebraic expression and identities
• Factorisation 
• Exponents and Powers
• Direct and Indirect Proportion
• Comparing Quantities

 

Linear Equations in one Variable

 

• An algebraic equation is an equality involving variables and an equality sign. The expression on the left of the equality sign is the Left Hand Side (LHS). The expression on the right of the equality sign is the Right Hand Side (RHS).
• An algebraic equation involving only one variable with its highest power 1 is called a linear equation.
• The values of the expressions on the LHS and RHS are equal. This happens to be true only for certain values of the variable. These values are the solutions of the equation.

 

Example

1. Solve the equation: \[\frac{\mathbf{0}\mathbf{.5}\left( \mathbf{z-0}\mathbf{.4} \right)}{\mathbf{3}\mathbf{.5}}\mathbf{-}\frac{\mathbf{0}\mathbf{.6}\left( \mathbf{z-2}\mathbf{.7} \right)}{\mathbf{4}\mathbf{.2}}\mathbf{=z+6}\mathbf{.1}\]

(a) \[-\frac{202}{35}\]    (b) \[\frac{202}{35}\]

(c) \[\frac{35}{202}\]      (d) \[-\frac{35}{202}\]

(e) None of these

Answer: (a)

Explanation: \[\frac{5\left( z-0.4 \right)}{35}-\frac{6\left( z-2.7 \right)}{42}=z+6.1\Rightarrow \frac{30z-12-30z+81}{210}=z+6.1\Rightarrow \frac{69}{210}=z+6.1\Rightarrow z=-\frac{202}{35}\]

 

2. David cuts a bread into two equal pieces and cuts one half into smaller pieces of equal size. Each of the small pieces is twenty gram in weight. If he has seven pieces of the bread all with him, how heavy is the original cake.

(a) 120 gm                     (b) 180 gm

(c) 300 gm                     (d) 240 gm

(e) None of these

Answer: (d)

Explanation: There are total of seven pieces, so number of smaller pieces is six.

Weight of each smaller piece is 20 gm

Therefore, weight of six such pieces is \[6\times ~20=120\text{ }gm\]

Hence the total weight of original cake \[=2\times ~120=240\text{ }gm\]

 

Algebraic expressions and identities

 

• The expression that contains only one term is called a monomial.

          For example,  \[3{{x}^{2}},8xy,\,-6z,\,9x{{y}^{2}},\,2x,\,-3,\,22qrs,\,\,\] etc. are the monomials.

• The expression that contains two terms is called a binomial.

          For example, \[4a+5b,\,3l-8m,\,2m+7,3-7{{x}^{2}}y,\,4{{x}^{2}}-{{z}^{2}},\] etc. are the binomials.

• The expression that contains two terms is called a trinomial.

          For example, \[a+b+c,\text{ }2x+3y-5z,{{x}^{2}}{{y}^{2}}z-{{x}^{3}}{{y}^{2}}z+1,\] etc. are the trinomials.

• An expression containing, one or more terms with non-zero coefficient (with variables having non-negative exponents) is called a polynomial. A polynomial may contain any number of terms, one or more than one.

          For example, \[6xy,\,\,8{{x}^{2}}yz-7,\,\,5x+9y+8z,\] etc. are the polynomials.

• The terms which contain similar variables having same powers are called like terms. Coefficients of like terms need not be the same.
• The terms which contain different variables are called unlike terms.
• Only like terms can be added or subtracted.
• Some standard algebraic identities are as follows:

         \[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]

         \[{{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}\]

         \[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\]

         \[\left( x+a \right)\left( x+b \right)={{x}^{2}}+\left( a+b \right)x+ab\]

 

Example

1. The product of \[\left( \mathbf{-3}{{\mathbf{x}}^{\mathbf{2}}}\mathbf{-x+7} \right)\] and \[\left( \mathbf{3-2x+}{{\mathbf{x}}^{\mathbf{2}}} \right)\] is____.

(a) \[-3{{x}^{4}}+5{{x}^{3}}-17x+21\]                    

(b) \[{{x}^{5}}+24{{x}^{4}}+5{{x}^{2}}+x+21\]

(c) \[8{{x}^{5}}+{{x}^{4}}-12+7x+1\]        

(d) \[3{{x}^{5}}-4{{x}^{4}}+1{{x}^{3}}-5{{x}^{2}}-7x+2\]

(e) None of these

Answer: (a)

Explanation: \[\left( 2{{x}^{2}}-5{{x}^{2}}-x+7 \right)\left( 3-2x+{{x}^{2}} \right)\]

= \[-9{{x}^{2}}+6{{x}^{3}}-3{{x}^{4}}-3x+2{{x}^{2}}-{{x}^{3}}+21-14x+7{{x}^{2}}\]

= \[3{{x}^{4}}+5{{x}^{3}}-17x+21\]

 

Factorisation

 

• An expression is said to be factorise when an expression is written as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions.
• A factor which cannot be expressed further as a product of factors is called an irreducible factor.
•  To factorise an algebraic expression, the above mentioned algebraic identities can be used.
• To factorise an algebraic expression of the type \[{{x}^{2}}~+\text{ }px+q,\] spilt p and q in such a way that \[ab=q\] and \[a+b=p\]. Then, the expression becomes

\[{{x}^{2}}~+\left( a+b \right)x+ab\]

\[={{x}^{2}}~+\text{ }ax+bx+ab\]

\[=x\left( x+a \right)+b\left( x+a \right)\]

\[=\left( x+a \right)\left( x+b \right)\]

Here \[\left( x+a \right)\] and \[\left( x+b \right)\] are the required factors.

 

Example

 

1. Factorise the given polynomial: \[\mathbf{36}{{\mathbf{u}}^{\mathbf{2}}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{25}}{{\mathbf{v}}^{\mathbf{2}}}\mathbf{+25}{{\mathbf{w}}^{\mathbf{2}}}\mathbf{-}\frac{\mathbf{12}}{\mathbf{5}}\mathbf{uv-2vw+60wu}\]

(a) \[{{\left( 6u-\frac{v}{5}-5w \right)}^{2}}\]    

(b) \[{{\left( 6u+\frac{v}{5}-5w \right)}^{2}}\]

(c) \[{{\left( 6u-\frac{v}{5}+5w \right)}^{2}}\]   

(d) \[{{\left( 6u+\frac{v}{5}+5w \right)}^{2}}\]

(e) None of these

Answer: (c)

 

2. Simplify the expression \[\mathbf{39}{{\mathbf{y}}^{\mathbf{2}}}\left( \mathbf{50}{{\mathbf{y}}^{\mathbf{2}}}\mathbf{-98} \right)\mathbf{\div 26}{{\mathbf{y}}^{\mathbf{2}}}\left( \mathbf{5y+7} \right)\].

(a) \[3\left( 5y+8 \right)\]                        

(b) \[15y+21\]

(c) \[15y-21\]                             

(d) \[18y-21\]

(e) None of these

Answer: (c)

 

 

3. Simplify: \[\left( \mathbf{2x}+\mathbf{3y} \right)\left( \mathbf{2x}-\mathbf{3y} \right).\]

(a) \[{{x}^{2}}+3{{y}^{2}}\]                       

(b) \[2{{x}^{2}}+3{{y}^{2}}\]

(c) \[4{{x}^{2}}-9{{y}^{2}}\]                        

(d) \[{{x}^{2}}-9{{y}^{2}}\]

(e) None of these

Answer: (c)

Explanation: \[\left( 2x+3y \right)\left( 2x-3y \right)\left( 2x-3y \right)={{\left( 2x \right)}^{2}}-{{\left( 3y \right)}^{2}}=4{{x}^{2}}-9\,{{y}^{2}}\]

 

Exponents and Powers

 

• Exponents and powers are used to express large and small numbers in standard form. Very small numbers can be expressed in standard form using negative exponents. Similarly very large numbers can be expressed in standard form using positive exponents.
• Numbers with positive or negative exponents obey the following rules:

(a) \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]                

(b) \[{{a}^{m}}\div {{a}^{n}}={{a}^{m-n}}\]

(c) \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\]              

(d) \[{{a}^{m}}\times {{b}^{m}}={{\left( ab \right)}^{m}}\]

(e) \[{{a}^{0}}=1\]                 

(f) \[\frac{{{a}^{m}}}{{{b}^{m}}}={{\left( \frac{a}{b} \right)}^{m}}\]

 

Example

1. Find the value of x such that \[{{\left( \frac{\mathbf{64}}{\mathbf{125}} \right)}^{\mathbf{2}}}{{\left( \frac{\mathbf{4}}{\mathbf{5}} \right)}^{\mathbf{4}}}{{\left( \frac{\mathbf{16}}{\mathbf{25}} \right)}^{\mathbf{2x+1}}}\mathbf{=}{{\left( \frac{\mathbf{256}}{\mathbf{625}} \right)}^{\mathbf{3x}}}\mathbf{.}\]

(a) \[\frac{3}{2}\]           (b) \[\frac{2}{3}\]

(c) \[\frac{1}{3}\]           (d) \[\frac{1}{2}\]

(e) None of these

Answer: (a)

 

Direct and Indirect Proportion

 

• Two quantities x and y are said to be in direct proportion If they increase or decrease together. Mathematically, two numbers x and y are said to be in direct proportion if, x/y = k or \[{{x}_{1}}/{{x}_{1}}\] = \[{{x}_{2}}/{{x}_{2}}\] where \[{{y}_{1}}\] and \[{{y}_{2}}\] are the values of y corresponding to the values \[{{x}_{1}}\]and\[{{x}_{2}}\].

 

• Two quantities x and y are said to be in inverse or indirect proportion if an increase in x causes a proportional decrease in y. Mathematically, two numbers x and y are said to be in inverse or indirect proportion if, \[xy=k\] or \[{{x}_{1}}{{y}_{1}}={{x}_{2}}{{y}_{2}}\] where \[{{y}_{1}}\] and \[{{y}_{2}}\] are the values of y corresponding to the values \[{{x}_{1}}\] and \[{{x}_{2}}\].

 

Example

1. If women or 3 men earn Rs. 960 in a day, then the earning of 11 women and 7 men in a day will be:

(a) Rs. 4880                   (b) Rs. 2200

(c) Rs. 1860                   (d) Rs. 1480

(e) None of these

Answer: (a)

Explanation: One day earning of 4 women or 3 men = Rs. 960

Therefore, one day earning of 1 women or 1 men =\[\frac{960}{4}\] or Rs. \[\frac{960}{3}\]

One day earning of 11 women or 7 men = Rs. \[\frac{960}{4}\times 11\] or Rs. \[\frac{960}{3}\times 7\]

= Rs. 2640 or Rs. 2240

Therefore, total earning of 11 women and 7 men for one day is Rs 4880.

 

2. A university has its own hostel for its students. 8t provides fooding and lodging to the students. Due to festive season some of the students are left for their home and 100 students stays in the hostel. There is a food provision for 20 days for these students. How long the food will last if 25 more students decide to stay back in the hostel?

(a) 12 days                    (b) 13 days

(c) 16 days                     (d) 14 days

(e) None of these

Answer: (c)

Explanation: Initially the number of students = 100.

Provision for food = 20 days.

Finally number of students = 125.

100x20

Number of days food will last \[=\frac{100\times 20}{125}~=16\text{ }days.\]

 

Comparing Quantities

• A reduction given on marked price is called discount. It can be calculated as:

         Discount = Marked Price - Sale Price (When M.P. and S.P. is given)

         Discount = Discount % of Marked Price (When discount percentage is given)

• Overhead expenses: The additional expenses made after buying an article are included in the cost price and are known as overhead expenses. Thus, cost price of an article is given by,

         Cost Price (C.P.) = Buying price + Overhead expenses

• The tax charged by the government on the sale of an item is called sales tax. It is added to the Bill Amount.

         Sales tax = Tax% of Bill Amount

• Value Added Tax (VAT): It is a form of consumption tax. From the perspective of the buyer, it is a tax on the purchase price. From that of the seller, it is a tax only on the value added to a product.
• When S.P. > C.P., seller gets profit which can be calculated as:

         Profit = S.P. - C.P.

• Profit percent is given by the formula, Profile % \[=\left( Profits/C.P \right)\times 100\]
• When S.P.  < C.P. seller is in loss which can be calculated as

         Loss = C.P. - S.P

• Loss percent is given by the formula, Loss% \[=\left( Loss/C.P \right)\times 100\]
• Simple interest is calculated as follows:

         SI = PRT/100 where P is principal, R is rate of interest and T is time,

• Compound interest is the interest calculated on the previous year's amount. It can be calculated as:

          \[C.I.=A-P\]

          Here, \[A=P{{\left( 1+\frac{r}{100} \right)}^{n}}\]

          (P = principal, r = rate of interest and n = time period)

 

Example

1. A landlord has a large piece of agricultural land which he wants to sell. James wants to buy the land and buys it for Rs. 400000. After some time he was in need of money and wanted to sell that land piece. He sells one third of the land at the loss of 20% and two fifth at the gain of 25%. At what price he must sell the remaining land so that he can make the overall profit of 10% on the whole transaction?

(a) \[Rs\frac{80000}{3}\]                       

(b) \[Rs\frac{400000}{3}\]

(c) \[Rs\frac{320000}{3}\]                      

(d) \[Rs\frac{920000}{3}\]

(e) None of these

Answer: (b)

Explanation: Cost price of entire land = Rs. 400000

Proposed profit on the whole land = 10% of 400000 = Rs. 40000

Proposed selling price of whole land = Rs. 440000

Cost price of one third land = Rs. \[\frac{400000}{3}\]

Loss on that land = 20% of \[Rs\frac{400000}{3}=Rs\frac{80000}{3}\]

Selling price of that land = \[\frac{400000}{3}-\frac{80000}{3}=Rs\frac{320000}{3}\]

Cost price of two fifth of the land = \[\frac{2}{5}\times 400000=Rs160000\]

Gain on the transaction \[=25%\]of \[160000=Rs.\text{ }40000\]

Selling price of that land \[=Rs.\left( 160000+40000 \right)=Rs.\text{ }200000\]

Total selling price of the land = \[\frac{320000}{3}+200000=Rs\frac{920000}{3}\]

Selling price of the remaining land = Rs. \[(440000-\frac{920000}{3})=Rs\frac{400000}{3}\]

 

2. If the cost price of 20 greeting cards is equal to the selling price of 16 greeting cards, find the gain or loss percent.

(a) 20%                                     (b) 25%

(c) 30%                                     (d) 40%

(e) None of these

Answer: (b)

Explanation: Let cost price of each card be\[Rs\text{ }x.\].

Then C. P. of 16 greeting cards \[=Rs.\text{ }16x\]

S. P. of 16 greeting cards = C.P. of 20 cards \[=Rs.\text{ }20x\]

Therefore, gain \[=S.P.-C.P.=\text{ }Rs\left( 20-16 \right)x=Rs.\text{ }4x\]

% gain = \[\frac{4x}{16x}\times 100=25%\]