Method of Cross Multiplication

Q.

Question 97

Solve the following:

The age of A is five years more than that of B. 5 years ago, the ratio of their ages was 3:2. Find their present ages.

Q.

Question 106

Solve the following

Kusum buys some chocolates at the rate of Rs. 10 per chocolate. She also buys an equal number of candies at the rate of Rs. 5 per candy. She makes a 20% profit on chocolates and 8% profit on candies. At the end of the day, all chocolates and candies are sold out and her profit is Rs. 240. Find the number of chocolates purchased.

Q. Question 3
For what values of a and b will the following pair of linear equations have infinitely many solutions?
x + 2y = 1
(a - b)x + (a + b)y = a + b - 2

Q. The total cost of 10 erasers and 5 sharpner is at least RS.65.The cost of each eraser cannot exceed Rs.4.Find the minimum cost of each sharpner.

Q. A boat covers 32 km upstream and 36 km downstream in 7 hours. Also, it covers 40 km upstream and 48 km downstream in 9 hours. Speed of stream is .

1. 6 km/hr
2. 4 km/hr
3. 2 km/hr

Q. A two-digit number is obtained by either multiplying the sum of the digits by $8$ and then subtracting $5$ or by multiplying the difference of the digits by $16$ and then adding $3$. Find the number.

1. $100$
2. $22$
3. $76$
4. $83$

Q.

If the cost of $1$ pen is ₨ $5$ and the cost of $1$ pencil is ₨ $2$. Find the cost of $\mathit{x}$ pens and $\mathit{y}$ pencils together.

Q. Two numbers are in the ratio $3:4$. If $4$ is added to each, the ratio becomes $4:5$. Find the numbers.

Q. Question 2 (v)
Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:
A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs 27 for a book kept for seven days, while Susy paid Rs 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day

Q. A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student $A$ takes food for $20$ days she has to pay Rs. $1000$ as hostel charges whereas a student $B$, who takes food for $26$ days, pays Rs. $1180$ as hostel charges. Then the fixed charges for both $A$ and $B$ must be Rs. $400$.

1. True
2. False

Q.

The cost of $4$ pens and $4$ pencil boxes is $₹100$. Three times the cost of a pen is $₹15$ more than the cost of a pencil box. Form the pair of linear equations for the above situation. Find the cost of a pen and a pencil box.

Q. Vijay had some bananas, and he divided them into two lots $A$ and $B$. He sold the first lot at the rate of Rs. $2$ for $3$ bananas and the second lot at the rate of Rs. $1$ per banana, and got a total of Rs. $400$. If he had sold the first lot at the rate of Rs. $1$ per banana, and the second lot at the rate of Rs. $4$ for $5$ bananas, his total collection would have been Rs. $460$. Find the total number of bananas he had.

1. $200$
2. $500$
3. $600$
4. $900$

Q.

When using cross multiplication method for solving

$a_{1}x$ + $b_{1}y$ + $c_1$ = 0

$a_{2}x$ + $b_{2}y$ + $c_2$ = 0

y is given by $\frac{c_1{a}_2-c_2{a}_1}{a_1{b}_2-a_2{b}_1}$

1. True

2. False

Q.

Solve the pair of linear equations using cross-multiplication method.

$\frac{5}{(x-2)}=\frac{4}{(1-y)}$

$26x+3y+4=0$

1. $x=2,y=\frac{1}{3}$

2. $x=-2,y=\frac{1}{3}$

3. $x=-\frac{1}{2},y=3$

4. $x=2,y=3$

Q.

A storekeeper wants to mix two types of flour to get $300\mathrm{pounds}$, so he can sell it by $\$2.50$ per pound.

If he uses flour worth $\$2.40$ a pound with another flour worth $\$3.00$ a pound, how many pounds of each does he use?

Q. In a competitive examination, one mark is awarded for each correct answer while $\frac{1}{2}$ mark is deducted for every wrong answer. Jayanti answered $120$ questions and got $90$ marks. How many questions did she answer correctly?

1. $70$
2. $66$
3. $125$
4. $100$

Q. A shopkeeper buys a number of books for Rs. 80. If he had bought 4 more books for the same amount, each book would have cost Rs. 1 less. How many books did he buy? [4 MARKS]

Q. Two numbers are in the ratio $5:6$. If $8$ is subtracted from each of the numbers, the ratio becomes $4:5$. Find the numbers.

1. $35,21$
2. $67,32$
3. $40,48$
4. $50,25$

Q.

A boat covers a distance of 32 km upstream and 36 km downstream in 7 hours. It also covers a distance of 40 km upstream and 48 km downstream in 9 hours. Find the speed of the boat in still water and that of the stream.

1. 10 km/hr, 2 km/hr

2. 14 km/hr, 5 km/hr

3. 6 km/hr, 4 km/hr

4. 44 km/hr, 8 km/hr

Q. A boat covers 32 km upstream and 36 km downstream in 7 hours. Also, it covers 40 km upstream and 48 km downstream in 9 hours. Find the speed of the boat in still water and that of the stream.

1. 44 km/hr, 8 km/hr
2. 6 km/hr, 4 km/hr
3. 14 km/hr, 5 km/hr
4. 10 km/hr, 2 km/hr

Q.

Solution for the given system of equation is

$\frac{x+y-8}{2}$ = $\frac{x+2y-14}{8}$ = $\frac{3x+y-12}{11}$

1. (1, -1)

2. (2, 6)

3. (0, 1)

4. (2, 2)

Q.

Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.

1. 6, 4

2. 4, 6

3. 8, 12

4. None of these

Q. A fraction becomes $4$ when $1$ is added to both the numerator and denominator and it becomes $7$ when 1 is subtracted from both. The numerator of the given fraction is?

1. $12$
2. $13$
3. $15$
4. $17$

Q.

Solve the following pair of equations and choose the correct answer.
2x + y = 7
3x + 2y = 12

1. (2, 3)

2. (3, 2)

3. (1, 0)

4. (-3, 2)

Q.

For the given pair of linear equations,
$a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$.

The values of x and y are ( ___, ___ ).

1. $\frac{(b_2{c}_2-b_1{c}_1)}{(a_1{b}_2-a_2{b}_1)},\frac{(c_2{a}_2-c_1{a}_1)}{(a_1{b}_2-a_2{b}_1)}$

2. $\frac{(b_2{c}_1-b_1{c}_2)}{(a_1{b}_2-a_2{b}_1)},\frac{(c_1{a}_1-c_2{a}_2)}{(a_1{b}_2-a_2{b}_1)}$

3. $\frac{(b_1{c}_2-b_2{c}_1)}{(a_1{b}_2-a_2{b}_1)},\frac{(c_1{a}_2-c_2{a}_1)}{(a_1{b}_2-a_2{b}_1)}$

4. $Noneoftheabove$

Q. If the numerator of a certain fraction is increased by 5 and the denominator is decreased by 1, the resulting fraction is $\frac{8}{3}$. However, if the numerator of the original fraction is doubled and the denominator is increased by 7, the resulting fractions is $\frac{6}{11}$. Find the original fraction.

1. $\frac{3}{4}$
2. $\frac{1}{4}$
3. $\frac{1}{2}$
4. $\frac{1}{3}$

Q. For the given pair of linear equations,
$a_{1}x+b_{1}y+c_1=0$ and
$a_{2}x+b_{2}y+c_2=0$.

The values of x and y are ( __ , __ ).
Fill in the blanks.

1. $\frac{(b_1{c}_2-b_2{c}_1)}{(a_1{b}_2-a_2{b}_1)},\frac{(c_1{a}_2-c_2{a}_1)}{(a_1{b}_2-a_2{b}_1)}$
2. $\frac{(b_2{c}_2-b_1{c}_1)}{(a_1{b}_2-a_2{b}_1)},\frac{(c_2{a}_2-c_1{a}_1)}{(a_1{b}_2-a_2{b}_1)}$
3. $\frac{(b_2{c}_1-b_1{c}_2)}{(a_1{b}_2-a_2{b}_1)},\frac{(c_1{a}_1-c_2{a}_2)}{(a_1{b}_2-a_2{b}_1)}$
4. $\frac{(b_1{a}_2-b_2{c}_1)}{(a_1{b}_2-a_2{b}_1)},\frac{(c_1{a}_2-c_2{a}_1)}{(a_1{b}_2-a_2{b}_1)}$

Q.

Express the following linear equations in the form $ax+by+c=0$ and indicate the values of a, b and c in each case:

(i) $-x+4y=12$ (ii) $x-\frac{y}{4}-5=0$ (iii) $2x+3y=9.35$

Q.

Identify the correct diagram related to cross-multiplication method for following pair of equations.

$2x+3y=44$

$3x+5y=76$

1. 

2. 

3. 

4. 

Q. Question 2 (i)
For which values of ‘a’ and ‘b’ does the following pair of linear equations have an infinite number of solutions?
2x + 3y =7
(a - b)x + (a + b)y = 3a +b - 2