This problem involves setting up and solving a system of linear equations. We will define variables for the original prices of a pen and a pencil. Then, we will translate the given conditions into two equations. After solving these equations to find the original prices, we will calculate the final required value.
Let the original price of one pen be \(P\) (in ₹) and the original price of one pencil be \(C\) (in ₹).
From the first statement, "The price of 2 pens and 19 pencils is ₹118", we can form the first equation:
\[ 2P + 19C = 118 \quad \text{(Equation 1)} \]From the second statement, "If the price of one pen decreases by ₹5 and the price of one pencil increases by ₹4, then the price of 5 pens and 8 pencils becomes ₹144".
The new price of one pen is \(P - 5\).
The new price of one pencil is \(C + 4\).
So, the second equation is:
\[ 5(P - 5) + 8(C + 4) = 144 \]\[ 5P - 25 + 8C + 32 = 144 \]\[ 5P + 8C + 7 = 144 \]\[ 5P + 8C = 144 - 7 \]\[ 5P + 8C = 137 \quad \text{(Equation 2)} \]Now we have a system of two linear equations:
Equation 1: \(2P + 19C = 118\)
Equation 2: \(5P + 8C = 137\)
To solve for \(P\) and \(C\), we can use the elimination method. Multiply Equation 1 by 5 and Equation 2 by 2:
Multiply Equation 1 by 5:
\[ 5(2P + 19C) = 5(118) \]\[ 10P + 95C = 590 \quad \text{(Equation 3)} \]Multiply Equation 2 by 2:
\[ 2(5P + 8C) = 2(137) \]\[ 10P + 16C = 274 \quad \text{(Equation 4)} \]Subtract Equation 4 from Equation 3:
\[ (10P + 95C) - (10P + 16C) = 590 - 274 \]\[ 10P + 95C - 10P - 16C = 316 \]\[ 79C = 316 \]\[ C = \frac{316}{79} \]\[ C = 4 \]Substitute the value of \(C = 4\) into Equation 1:
\[ 2P + 19(4) = 118 \]\[ 2P + 76 = 118 \]\[ 2P = 118 - 76 \]\[ 2P = 42 \]\[ P = \frac{42}{2} \]\[ P = 21 \]So, the original price of one pen is ₹21 and the original price of one pencil is ₹4.
We need to find the actual price of 14 pens and 6 pencils. This means using the original prices \(P\) and \(C\).
Required price \( = 14P + 6C \)
Required price \( = 14(21) + 6(4) \)
Required price \( = 294 + 24 \)
Required price \( = 318 \)
A) ₹318 is the correct answer, as calculated from the original prices of pens and pencils.
This problem involves the relationship between the Highest Common Factor (HCF) and Least Common Multiple (LCM) of two numbers. A fundamental property states that the product of two numbers is equal to the product of their HCF and LCM. We can use this property to find the unknown number.
For any two positive integers, the product of the numbers is equal to the product of their HCF and LCM.
\[ \text{Number 1} \times \text{Number 2} = \text{HCF} \times \text{LCM} \]To find \( x \), divide the product of HCF and LCM by the given number.
\[ x = \frac{91 \times 546}{182} \]Notice that \( 182 = 2 \times 91 \). So, we can simplify the expression:
\[ x = \frac{91 \times 546}{2 \times 91} \]\[ x = \frac{546}{2} \]\[ x = 273 \]The other number is 273.
A) 273 is the correct answer. The calculation shows that the other number is 273, which satisfies the property that the product of the two numbers (182 and 273) equals the product of their HCF (91) and LCM (546).
To find the original number, we need to understand that increasing a number by 60% means the new number is 100% (original) + 60% (increase) = 160% of the original number. We can set up an equation to solve for the original number.
Let the original number be \(x\).
When the number is increased by 60%, it means we are adding 60% of \(x\) to \(x\). Mathematically, this can be written as:
\[ x + 0.60x \]This sum is equal to 3550. So, we can form the equation:
\[ x + 0.60x = 3550 \]Combine the terms on the left side:
\[ 1.60x = 3550 \]To find \(x\), divide both sides by 1.60:
\[ x = \frac{3550}{1.60} \]Perform the division:
\[ x = 2218.75 \]Correct Option: A) 2218.75
This problem involves calculating the total profit percentage of a dishonest trader. The trader makes a profit in two ways: by claiming a profit margin on the selling price and by using a faulty weight. We need to combine these two profit sources to find the overall profit percentage.
C) 67%
To solve this problem, we need to evaluate each square root term individually and then perform the addition and subtraction operations in the given order.
We know that \(20 \times 20 = 400\).
\[ \sqrt{400} = 20 \]First, convert the decimal to a fraction:
\[ 0.0225 = \frac{225}{10000} \]Now, find the square root:
\[ \sqrt{0.0225} = \sqrt{\frac{225}{10000}} = \frac{\sqrt{225}}{\sqrt{10000}} \]We know that \(15 \times 15 = 225\) and \(100 \times 100 = 10000\).
\[ \sqrt{0.0225} = \frac{15}{100} = 0.15 \]We know that \(3 \times 3 = 9\).
\[ \sqrt{9} = 3 \]The original expression is \(\sqrt{400} + \sqrt{0.0225} - \sqrt{9}\).
Substitute the calculated values:
\[ 20 + 0.15 - 3 \]Correct Option: D) 17.15
The problem asks for the largest number that divides 527 and 632, leaving remainders 7 and 8 respectively. This means that if we subtract the remainders from the original numbers, the resulting numbers will be perfectly divisible by the required largest number. Therefore, the required number is the Highest Common Factor (HCF) of these new numbers.
When 527 is divided by the number, the remainder is 7. So, \(527 - 7 = 520\) must be perfectly divisible by the number.
When 632 is divided by the number, the remainder is 8. So, \(632 - 8 = 624\) must be perfectly divisible by the number.
We need to find the HCF of 520 and 624. We can use the prime factorization method or the Euclidean algorithm.
Using prime factorization:
The common prime factors are \(2\) and \(13\). The lowest power of \(2\) common to both is \(2^3\). The lowest power of \(13\) common to both is \(13^1\).
So, HCF \((520, 624) = 2^3 \times 13 = 8 \times 13 = 104\).
Alternatively, using the Euclidean algorithm:
The last non-zero remainder is 104, so the HCF is 104.
The conditions are satisfied.
Correct Option: B) 104
To find A's share, we need to determine the ratio of the shares of A, B, and C. The problem states a relationship between their shares: 9 times A's share = 3 times B's share = 7 times C's share. We will use this relationship to find the ratio A:B:C and then distribute the total amount accordingly.
Let A's share be \(S_A\), B's share be \(S_B\), and C's share be \(S_C\).
According to the problem statement, we have the equation:
\[ 9 S_A = 3 S_B = 7 S_C \]To find the ratio \(S_A : S_B : S_C\), we can set each part of the equation equal to a constant, say \(k\):
\[ 9 S_A = k \implies S_A = \frac{k}{9} \]\[ 3 S_B = k \implies S_B = \frac{k}{3} \]\[ 7 S_C = k \implies S_C = \frac{k}{7} \]Now, the ratio \(S_A : S_B : S_C\) is:
\[ S_A : S_B : S_C = \frac{k}{9} : \frac{k}{3} : \frac{k}{7} \]\[ S_A : S_B : S_C = \frac{1}{9} : \frac{1}{3} : \frac{1}{7} \]To convert this ratio into whole numbers, we find the Least Common Multiple (LCM) of the denominators (9, 3, 7). The LCM of 9, 3, and 7 is \(9 \times 7 = 63\).
Multiply each part of the ratio by the LCM (63):
\[ S_A : S_B : S_C = \left(\frac{1}{9} \times 63\right) : \left(\frac{1}{3} \times 63\right) : \left(\frac{1}{7} \times 63\right) \]\[ S_A : S_B : S_C = 7 : 21 : 9 \]The sum of the ratio parts is \(7 + 21 + 9 = 37\).
The total amount to be divided is ₹74,000.
A's share can be calculated as the fraction of A's ratio part to the total ratio sum, multiplied by the total amount:
\[ S_A = \frac{7}{37} \times \text{₹}74,000 \]\[ S_A = 7 \times \frac{74,000}{37} \]\[ S_A = 7 \times 2,000 \]\[ S_A = \text{₹}14,000 \]Correct Option: C) ₹14,000
To simplify the given algebraic expression, we will first expand the products using the distributive property and then combine like terms. This involves multiplying terms inside parentheses by the terms outside and then adding or subtracting coefficients of terms with the same variable and exponent.
Correct Option: B) \(11x^2 - 3x - 2\)
To find the number of prime numbers between 466 and 476 (inclusive), we need to list all integers in this range and then check each one for primality. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
Therefore, there is only 1 prime number between 466 and 476 (both inclusive).
Correct Option: A) 1
To find the trader's profit, we need to calculate the total cost price, the number of plates remaining after breakage, and the total selling price of the remaining plates. The profit is then the difference between the total selling price and the total cost price.
D) ₹310.40 is the correct profit calculated.