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Order of operations requires multiplication before addition and subtraction, so 8 × 3 = 24 is computed first. Then 7 + 24 − 4 = 27. Working strictly left to right would give a different, incorrect result.
A number is divisible by 9 exactly when the sum of its digits is a multiple of 9. For 4,563 the digits sum to 4 + 5 + 6 + 3 = 18, which is divisible by 9. Hence 4,563 is divisible by 9.
A 15% reduction removes 0.15 × 800 = 120 taka from the original price. Subtracting this discount from 800 gives 680 taka. Equivalently, the sale price is 85% of 800, which is 680.
Writing 36 = 2² × 3² and 48 = 2⁴ × 3, the HCF takes the lowest power of each shared prime. This gives 2² × 3 = 12. So 12 is the largest number dividing both.
Dividing numerator and denominator by their HCF, 42, gives 84 ÷ 42 = 2 and 126 ÷ 42 = 3. The fraction in lowest terms is therefore 2/3. No common factor greater than 1 remains.
Since 0.375 has three decimal places, it equals 375/1000. Dividing numerator and denominator by 125 gives 3/8. Thus 0.375 = 3/8 in lowest terms.
When multiplying powers of the same base you add the exponents, giving 2⁷. Dividing by 2² then means subtracting exponents: 2⁷ ÷ 2² = 2⁵. So the expression simplifies to 2⁵ = 32.
The mean is the total of the values divided by how many there are. The sum is 4 + 9 + 11 + 16 = 40, and dividing by 4 gives 10. Hence the average is 10.
Simple interest is calculated as principal × rate × time, giving 5000 × 0.08 × 3. This equals 5,000 × 0.24 = 1,200 taka. The interest does not compound because it is simple interest.
The ratio 3 : 5 means the total is split into 3 + 5 = 8 equal parts. Each part is 96 ÷ 8 = 12, so the larger share is 5 × 12 = 60. The smaller number is 36, and 60 + 36 = 96 confirms the result.
Their combined rate is 1/12 + 1/18 = 3/36 + 2/36 = 5/36 of the job per day. The time needed is the reciprocal, 36/5 = 7.2 days. So together they finish in 7.2 days.
The average speed is distance divided by time, 240 ÷ 3 = 80 km/h. At this constant speed, the distance in 5 hours is 80 × 5 = 400 km. Speed remains constant, so distance is proportional to time.
The selling price after a 20% discount is 0.80 × 500 = 400 taka. A 25% profit means the selling price is 1.25 times the cost, so cost = 400 ÷ 1.25 = 320 taka. Checking, 320 × 1.25 = 400 confirms the answer.
Successive percentage changes multiply rather than add. After the two increases the population is 1.10 × 1.20 = 1.32 times the original, an increase of 32%. Simply adding 10% and 20% would overcount the compounding.
The digits of 2,498 sum to 2 + 4 + 9 + 8 = 23. The next multiple of 9 above 23 is 27, so 4 must be added to reach divisibility. Indeed 2,498 + 4 = 2,502, whose digits sum to 9.
Prime factorising gives 12 = 2² × 3, 15 = 3 × 5, and 20 = 2² × 5. The LCM takes the highest power of each prime: 2² × 3 × 5 = 60. So 60 is the smallest number divisible by all three.
Let milk be 5x and water be 3x litres. After adding 4 litres of water, 5x : (3x + 4) = 5 : 4, so 20x = 15x + 20, giving x = 4. Therefore milk is 5 × 4 = 20 litres, and water is 12 litres.
Dividing by a fraction is the same as multiplying by its reciprocal. So (3/4) ÷ (9/8) = (3/4) × (8/9) = 24/36, which simplifies to 2/3. Cancelling common factors confirms the result.
Multiplying 0.4 by 0.25 gives 0.4 × 0.25 = 0.100, which equals 0.1. The two factors together have three decimal places, so the product has three decimal places before trailing zeros are dropped.
Multiplying powers of the same base adds exponents: 2⁵ × 2³ = 2⁸. Then dividing by 2⁶ subtracts exponents: 2⁸ ÷ 2⁶ = 2². The expression therefore equals 4.
The total of the five numbers is 5 × 18 = 90, and the total of the remaining four is 4 × 16 = 64. The removed number is the difference, 90 − 64 = 26. Checking, the four remaining numbers average 16 as required.
The net filling rate is 1/6 − 1/9 = 3/18 − 2/18 = 1/18 of the tank per hour. The time to fill is the reciprocal, 18 hours. So with both open, the tank fills in 18 hours.
For equal distances at speeds a and b, the average speed is the harmonic mean 2ab/(a + b). Here it is 2 × 60 × 40 ÷ 100 = 4,800 ÷ 100 = 48 km/h. The arithmetic mean of 50 would be wrong because more time is spent at the slower speed.
Doubling means the interest equals the principal, so 8 years of interest equals one principal. To become four times the original, the interest must equal three principals, requiring 3 × 8 = 24 years, not 16. Careful reading shows the total is four times, so 24 years is correct.
A 20% profit on cost means the cost price of the mixture is 54 ÷ 1.20 = 45 taka per kg. By the rule of alligation, the ratio of the 40 taka to the 60 taka variety is (60 − 45) : (45 − 40) = 15 : 5 = 3 : 1. So the cheaper variety is used three times as much.
Place value depends on the position of a digit within the number. Reading from the right, the digits occupy units, tens, hundreds, thousands, ten-thousands, hundred-thousands, and millions. The 7 sits in the hundred-thousands place, so its value is 7 multiplied by 100,000, which equals 700,000.
A number is divisible by 6 exactly when it is divisible by both 2 and 3. Divisibility by 2 requires an even final digit, and divisibility by 3 requires the digit sum to be a multiple of 3. For 4,326 the last digit is even and the digit sum 4+3+2+6 = 15 is a multiple of 3, so the number is divisible by 6.
Successive percentage changes multiply rather than add. Increasing by 25% multiplies the number by 1.25, and decreasing by 20% multiplies by 0.8. Since 1.25 × 0.8 = 1.00, the final value equals the original, so there is no net change.
For any two positive integers, the product of the numbers equals the product of their HCF and LCM. Here HCF × LCM = 12 × 180 = 2,160, and dividing by the known number gives 2,160 ÷ 36 = 60. Checking, the HCF of 36 and 60 is 12 and their LCM is 180, which agrees with the given data.
Under simple interest the amount grows by a fixed sum each year, so the extra 3 years add 1,800 − 1,500 = 300 taka, giving 100 taka of interest per year. Over 5 years the interest is 500 taka, so the principal is 1,500 − 500 = 1,000 taka. Checking, 8 years of interest gives 800 taka and an amount of 1,800 taka, as stated.
Add 7 to both sides to obtain 3x = 21, then divide both sides by 3 to isolate x, giving x = 7. The operation is valid because adding the same number to both sides preserves equality.
Multiply each term of the first bracket by each term of the second: x^2 - 3x + 5x - 15. Combining the like terms -3x and 5x gives 2x, so the result is x^2 + 2x - 15.
The expression is a difference of two squares, since x^2 - 9 = x^2 - 3^2. Such an expression factorises as (x - 3)(x + 3), and neither factor can be broken down further over the integers.
The gradient equals the change in y divided by the change in x, that is (11 - 3)/(6 - 2). This gives 8/4 = 2, so the line rises two units for every one unit to the right.
Multiply the numerical coefficients, 2 times 4, to get 8. For the powers of a, add the indices because the bases are the same: a^3 times a^5 equals a^8.
Substitute x = -2 into each term: 2(-2)^2 - 3(-2) + 1. Since (-2)^2 = 4, this becomes 8 + 6 + 1, which sums to 15.
Add 5 to both sides to get 4x < 16, then divide by the positive number 4, which keeps the inequality direction unchanged. This gives x < 4.
The logarithm asks for the exponent to which the base must be raised to give the argument. Since 2^5 = 32, the value of log base 2 of 32 is 5.
Here the first term a = 3 and the common difference d = 4, so the 20th term is 3 + 19(4) = 79. The sum is n/2 times the first plus last term, that is 10(3 + 79) = 820.
A quadratic with roots p and q can be written as x^2 - (p + q)x + pq = 0. Here the sum of the roots is 1 and the product is -12, giving x^2 - x - 12 = 0.
Adding the two equations eliminates y, giving 3x = 12, so x = 4. Substituting back into x - y = 1 gives y = 3, and both equations are then satisfied.
Factorise the numerator as (x - 2)(x + 2) and the denominator as (x + 3)(x - 2). Cancelling the common factor (x - 2) leaves (x + 2)/(x + 3), valid provided x is not 2 or -3.
The nth term of a geometric sequence is ar^(n-1), where a is the first term and r the common ratio. Here the 4th term is 5 times 3^3, that is 5 times 27, which equals 135.
Write 8 as 2^3 so that the right side becomes 2^(3x-3). Equating the exponents gives x + 1 = 3x - 3, which rearranges to 4 = 2x, so x = 2.
Two numbers whose product is 5 and whose sum is 6 are 1 and 5, so the expression factorises as (x - 1)(x - 5) = 0. Setting each factor to zero gives x = 1 or x = 5.
Write y = 3x - 6 and solve for x: add 6 to both sides to get y + 6 = 3x, then divide by 3. Swapping the variable names gives the inverse f^-1(x) = (x + 6)/3.
By the product rule, the sum of logarithms with the same base equals the logarithm of the product. Since 6 times 1.5 = 9 and 3^2 = 9, the expression equals log base 3 of 9, which is 2.
Equal roots occur when the discriminant b^2 - 4ac is zero. Here this gives k^2 - 36 = 0, so k^2 = 36 and k = 6 or k = -6.
Factorise as (x - 5)(x + 1) < 0, whose critical values are x = 5 and x = -1. The parabola opens upward, so it lies below the x-axis strictly between the roots, giving -1 < x < 5.
The general term is C(5, r) times 2^(5-r) times x^r. For x^3 we take r = 3, giving C(5,3) times 2^2, which is 10 times 4 = 40.