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Debt of country F = $250 million

Percentage interest = 3%

∴ Interest paid per year = 3% of 250 = 3/100 × 250 = $ 7.5 million = $7500000$

Debt of country A = $600 million

Debt of country F = $250 million

∴ Required ratio = 600 : 250 = 12 : 5

Debt of country A = $600 million

Debt of country B = $800 million

⇒ Difference in debt = 800 - 600 = 200

∴ Debt of country A is lesser than that of country B by = 200/800 × 100 = 25%

The company F has the lowest debt of $250 million, while the second lowest is company D of $500 million

∴ Company D has the second lowest debt

Given:

Simplify: \(\frac{3}{2} \times \frac{11}{5} \div \left(\frac{25}{44} \times \frac{11}{5}\right) \div \frac{33}{15}\)

Calculation:

Step 1: Simplify the expression inside the parentheses:

\(\frac{25}{44} \times \frac{11}{5} = \frac{25 \times 11}{44 \times 5} = \frac{275}{220} = \frac{5}{4}\)

Step 2: The expression now becomes:

\(\frac{3}{2} \times \frac{11}{5} \div \frac{5}{4} \div \frac{33}{15}\)

Step 3: Simplify the division operations:

\(\frac{11}{5} \div \frac{5}{4} = \frac{11}{5} \times \frac{4}{5} = \frac{44}{25}\)

Step 4: The expression now becomes:

\(\frac{3}{2} \times \frac{44}{25} \div \frac{33}{15}\)

Step 5: Simplify the next division:

\(\frac{44}{25} \div \frac{33}{15} = \frac{44}{25} \times \frac{15}{33} = \frac{44 \times 15}{25 \times 33} = \frac{660}{825} = \frac{4}{5}\)

Step 6: Final multiplication:

\(\frac{3}{2} \times \frac{4}{5} = \frac{12}{10} = \frac{6}{5}\)

∴ The simplified value is \(\frac{6}{5}\).

Given:

Area of the square that is bent to form a circle = 169 cm2 

Formulas used:

The perimeter of the square with side a = 4a

Area of the square with side a = a²

The perimeter of the circle with radius r = 2πr

The area of the circle with radius r = πr²

Calculation:

Area of the square =  a2 

⇒  a2 = 169 

⇒ a = ± 13 

Side of the square = 13 cm 

When a wire is bent to form a square and then a circle, the perimeter of both the figures will be equal. 

4a = 2πr 

⇒ 4 × 13 = 2 × 22/7 × r 

⇒ r = (13 × 7)/ 11 

Area of the circle = πr2 

⇒ 22/7 × (13 × 7)/ 11 × (13 × 7)/ 11 

⇒ 2366/11 

⇒ 215.09 cm²

∴ Area of the circle = 215 cm² (nearest to a whole number)

Shortcut Trick

To find the LCM of fractions, calculate the LCM of the numerators divided by the HCF of the denominators.

Numerators: 2, 3, 4 → LCM = 12

Denominators: 5, 10, 15 → HCF = 5

Required LCM = 12/5

∴ The correct answer is 12/5.

Alternate Method

Given: Fractions are 2/5, 3/10, and 4/15.

Formula Used: LCM of (a/b, c/d, e/f) = LCM(a, c, e) ÷ HCF(b, d, f)

Step-by-step Calculation:

⇒ Numerators of the given fractions are 2, 3, and 4.

⇒ Find the Lowest Common Multiple (LCM) of 2, 3, and 4:

⇒ Multiples of 2: 2, 4, 6, 8, 10, 12, ...

⇒ Multiples of 3: 3, 6, 9, 12, ...

⇒ Multiples of 4: 4, 8, 12, ...

⇒ LCM(2, 3, 4) = 12

⇒ Denominators of the given fractions are 5, 10, and 15.

⇒ Find the Highest Common Factor (HCF) of 5, 10, and 15:

⇒ Factors of 5: 1, 5

⇒ Factors of 10: 1, 2, 5, 10

⇒ Factors of 15: 1, 3, 5, 15

⇒ HCF(5, 10, 15) = 5

⇒ Therefore, LCM of fractions = 12/5

∴ The correct answer is 12/5.

Additional Information

HCF of Fractions

To find the HCF of fractions, use the formula: HCF of Numerators / LCM of Denominators.

Relation between Two Numbers

For any two positive integers a and b, the product of the numbers is equal to the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b).

Co-prime Numbers

If two numbers are co-prime (their HCF is 1), then their LCM is simply the product of the two numbers.

Shortcut Trick

Convert the given fractional relation directly to a ratio: (3 × 5) : (4 × 8) = 15 : 32.

Total ratio parts = 15 + 32 = 47.

47 parts → 94 ⇒ 1 part → 2.

First part = 15 parts × 2 = 30.

∴ The correct answer is 30.

Alternate Method

Given:

Total value to be divided = 94.

Let the two parts be A and B.

Logic / Formula Used:

If a number N is divided in the ratio m : n, then First Part = N × [m ÷ (m + n)].

Calculations:

⇒ According to the question, (A ÷ 5) ÷ (B ÷ 8) = 3 ÷ 4

⇒ (A ÷ 5) × (8 ÷ B) = 3 ÷ 4

⇒ 8A ÷ 5B = 3 ÷ 4

⇒ A ÷ B = (3 × 5) ÷ (4 × 8)

⇒ A ÷ B = 15 ÷ 32

⇒ Ratio of A : B = 15 : 32

⇒ Sum of ratio terms = 15 + 32 = 47

⇒ First part A = 94 × (15 ÷ 47)

⇒ First part A = 2 × 15 = 30

∴ The correct answer is 30.

Additional Information

Dividing a Quantity in a Given Ratio

To divide a quantity N in the ratio a : b, the first part is calculated as (a ÷ (a + b)) × N and the second part as (b ÷ (a + b)) × N.

Compound Ratio

The compound ratio of two ratios a : b and c : d is given by (a × c) : (b × d). It involves multiplying the antecedent and consequent terms respectively to form a single combined ratio.

Proportion

If four numbers a, b, c, and d are in proportion (a : b = c : d), then the product of extremes equals the product of means (a × d = b × c).

Shortcut Trick

Required smallest number = LCM of divisors + Remainder

LCM(15, 20, 35) = 420

Required smallest number = 420 + 8 = 428

∴ The correct answer is 428.

Alternate Method

Given:

Divisors = 15, 20, and 35

Remainder in each case = 8

Formula Used:

Smallest Number = LCM(a, b, c) + R

Calculation:

⇒ Prime factorization of 15 = 3 × 5

⇒ Prime factorization of 20 = 2² × 5

⇒ Prime factorization of 35 = 5 × 7

⇒ LCM(15, 20, 35) = 2² × 3 × 5 × 7 = 420

⇒ Smallest number = 420 + 8 = 428

∴ The correct answer is 428.

Additional Information

LCM with Different Remainders

If a number when divided by a, b, c leaves remainders p, q, r respectively and (a − p) = (b − q) = (c − r) = k, the smallest such number is LCM(a, b, c) − k.

Relation between HCF and LCM

For any two numbers, Product of numbers = HCF × LCM.

Co-prime Numbers

Two numbers are co-prime if their HCF is 1. The LCM of two co-prime numbers is their product.

Given:

The curved surface area of a cylindrical object is 990 m2 and its volume is 4455 m3.

Formula used:

The curved surface area of the cylindrical = 2πrh

The volume of the cylindrical = πr²h 

Calculation:

The curved surface area of the cylindrical = 2πrh

⇒ 2πrh = 990

⇒ πrh = 495 →(1)

The volume of the cylindrical = πr2h 

⇒ r(πrh) = 4455

⇒ r = \(\dfrac{4455}{495}\) = 9 m 

From Eqn (1), We have

⇒ π× 9 × h = 495 

⇒ h = \(\dfrac{495 \times7}{22 \times 9}\) = 17.5 m

According to the question,

The ratio of the height and diameter = h : 2r

⇒ 17.5 : 18

⇒ 35 : 36

∴ The ratio of the height and diameter of this object is 35 : 36.