To find the LCM of decimal numbers, we first convert them into fractions. Then, we find the LCM of the numerators and the HCF of the denominators. The LCM of the fractions is given by the formula:
\[ \text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of denominators}} \]The numerators are 54 and 3.
\( \text{LCM}(54, 3) \)
Since 54 is a multiple of 3 (\(54 = 3 \times 18\)), the LCM is 54.
The denominators are 5 and 50.
\( \text{HCF}(5, 50) \)
Since 5 is a factor of 50 (\(50 = 5 \times 10\)), the HCF is 5.
Alternative Method:
The numbers become 1080 and 6.
\( \text{LCM}(1080, 6) \)
Since 1080 is a multiple of 6 (\(1080 = 6 \times 180\)), the LCM is 1080.
Since we had two decimal places, place the decimal point two places from the right in the LCM obtained.
\(1080 \rightarrow 10.80\)
So, the LCM is 10.80 or 10.8.
A) 10.8 is the correct LCM of 10.8 and 0.06.
Simplify the following.
3((\( \frac{6}{3} \))x² − 21x + 18) − 6(x² + 5x − 18)
To simplify the given algebraic expression, we will first distribute the constants into the parentheses, then combine like terms. This involves performing multiplication and then addition/subtraction of coefficients for terms with the same variable and exponent.
Correct Option: D) \(-93x + 162\)
To find the discount percentage for a "buy X get Y free" offer, we need to determine the number of free items received and the total number of items involved in the transaction. The discount percentage is calculated as the ratio of free items to the total items, multiplied by 100.
Correct Option: D) 37.5% is the correct discount percentage based on the calculation.
To find the average weight of the family, we need to sum the weights of all individual members and then divide this total sum by the number of members in the family.
Correct Option: D) 51.2 kg is the calculated average weight of the family.
This problem involves profit sharing in a business, which is directly proportional to the investment made by each partner. We will first determine the ratio of profits, which will give us the ratio of their investments. Then, using the total investment, we can find individual investments and their difference.
Calculate Kushal's share of profit:
Total profit = ₹16,000
Brijesh's share = ₹4,400
\[ \text{Kushal's share} = \text{Total profit} - \text{Brijesh's share} \\ \text{Kushal's share} = \text{₹}16,000 - \text{₹}4,400 = \text{₹}11,600 \]Determine the ratio of their profits (and thus investments):
The ratio of profits is equal to the ratio of their investments.
\[ \text{Kushal's profit} : \text{Brijesh's profit} = 11,600 : 4,400 \]\[ \text{Ratio} = \frac{11,600}{4,400} = \frac{116}{44} = \frac{29 \times 4}{11 \times 4} = \frac{29}{11} \]\[ \text{So, Kushal's investment} : \text{Brijesh's investment} = 29 : 11 \]Calculate the individual investments:
Let Kushal's investment be \(29x\) and Brijesh's investment be \(11x\).
Total investment = ₹75,100
\[ 29x + 11x = 75,100 \]\[ 40x = 75,100 \]\[ x = \frac{75,100}{40} = \frac{7510}{4} = 1877.5 \]\[ \text{Kushal's investment} = 29 \times 1877.5 = \text{₹}54,447.50 \]\[ \text{Brijesh's investment} = 11 \times 1877.5 = \text{₹}20,652.50 \]Find the difference between their investments:
\[ \text{Difference} = \text{Kushal's investment} - \text{Brijesh's investment} \]\[ \text{Difference} = \text{₹}54,447.50 - \text{₹}20,652.50 = \text{₹}33,795 \]Alternatively, the difference can be calculated as:\[ \text{Difference} = (29x - 11x) = 18x \]\[ \text{Difference} = 18 \times 1877.5 = \text{₹}33,795 \]D) ₹33,795
We need to find the measure of \(\angle AEB\) in \(\triangle ABC\). We are given that \(BD \perp AC\), \(\angle DBC = 72°\), and \(\angle CAE = 56°\). We will use the properties of angles in triangles, specifically the sum of angles in a triangle and the exterior angle theorem.
Analyze \(\triangle BDC\):
Given that \(BD \perp AC\), it means \(\angle BDC = 90°\).
In \(\triangle BDC\), the sum of angles is \(180°\):
\[ \angle DBC + \angle BCD + \angle BDC = 180° \]\[ 72° + \angle BCD + 90° = 180° \]\[ 162° + \angle BCD = 180° \]\[ \angle BCD = 180° - 162° \]\[ \angle BCD = 18° \]Identify \(\angle C\) of \(\triangle ABC\):
From the previous step, we found \(\angle BCD = 18°\). This is the same as \(\angle C\) in \(\triangle ABC\).
So, \(\angle C = 18°\).
Analyze \(\triangle AEC\):
We are given \(\angle CAE = 56°\) and we found \(\angle ACE = \angle C = 18°\).
The angle \(\angle AEB\) is an exterior angle to \(\triangle AEC\) at vertex E.
According to the exterior angle theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two opposite interior angles.
Thus,
\[ \angle AEB = \angle CAE + \angle ACE \]\[ \angle AEB = 56° + 18° \]\[ \angle AEB = 74° \]Correct Option: D) 74°
The problem asks us to find the greater of two consecutive natural numbers whose product is 240. We can solve this by setting up an algebraic equation or by testing the options provided.
We are looking for the greater of the two numbers. Let's assume each option is the greater number and find the smaller consecutive number, then check their product.
Correct Option: D) The greater of the two numbers is 16. The two consecutive natural numbers are 15 and 16, and their product is \(15 \times 16 = 240\).
The simple interest on a certain sum for 7 \( \frac{2}{3} \) years at the rate of 5.4% per annum is ₹4,347. What will be the amount payable on the same sum at the rate of 7.5% per annum simple interest in 4 \( \frac{4}{5} \) years?
We need to first find the principal sum using the given simple interest, time, and rate. Once the principal is known, we can calculate the amount payable (Principal + Simple Interest) for the second scenario with a different rate and time period.
Correct Option: B) ₹14,280
To find the mean proportional between two numbers, say \(a\) and \(b\), we need to find a number \(x\) such that the ratio of \(a\) to \(x\) is equal to the ratio of \(x\) to \(b\). This can be expressed as \(a:x = x:b\), which simplifies to \(x^2 = ab\), or \(x = \sqrt{ab}\).
Correct Option: B) 52
The problem asks us to calculate the total area Mahita needs to paint. The room is shaped like a cube, and she is painting the four walls. We need to find the area of one wall and then multiply it by four, since all walls of a cube are identical squares.
Identify the shape and dimensions: The room is a cube with a side length of 18.5 feet.
Determine the shape of each wall: In a cube, each wall is a square.
Calculate the area of one wall: The area of a square is given by the formula \( \text{Area} = \text{side} \times \text{side} \) or \( \text{side}^2 \).
Given side length \( s = 18.5 \text{ feet} \).
Area of one wall \( = (18.5 \text{ feet})^2 \)
\[ 18.5 \times 18.5 = 342.25 \text{ square feet} \]
Calculate the total area to be painted: Mahita is painting four walls. So, we multiply the area of one wall by 4.
Total area \( = 4 \times \text{Area of one wall} \)
\[ 4 \times 342.25 \text{ square feet} = 1369 \text{ square feet} \]
Thus, the total area Mahita needs to paint is 1369 square feet.
The calculation shows that the total area to be painted is 1369 square feet, which matches option D.