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In the base-ten place-value system each position is ten times the one to its right. Reading 4,725 from the right, the digits occupy units, tens, hundreds and thousands places. The 7 sits in the hundreds place, so its value is 7 × 100 = 700.
A prime has exactly two factors, 1 and itself. Testing divisibility, 51 = 3 × 17, 57 = 3 × 19 and 63 = 3 × 21, so all three are composite. The number 59 is not divisible by 2, 3, 5 or 7, and since 7² exceeds 59 no larger factor need be checked, so 59 is prime.
Prime factorise both numbers: 36 = 2² × 3² and 48 = 2⁴ × 3. The HCF takes each common prime to the lowest power present, giving 2² × 3 = 12. Note that 144 is the LCM, not the HCF.
Write 8 = 2³ and 12 = 2² × 3. The LCM uses each prime to its highest power, giving 2³ × 3 = 24. Equivalently, LCM = (8 × 12) ÷ HCF = 96 ÷ 4 = 24.
A terminating decimal with three places equals the digits over 1000: 0.375 = 375/1000. Dividing numerator and denominator by 125 gives 3/8, and checking 3 ÷ 8 = 0.375 confirms it.
Multiply as whole numbers: 6 × 4 = 24. The factors have one decimal place each, so the product needs two decimal places, giving 0.24. Estimating 0.6 × 0.4 as a bit less than 0.6 × 0.5 = 0.3 supports this.
A number is divisible by 9 exactly when the sum of its digits is a multiple of 9. For 4,527 the digit sum is 4 + 5 + 2 + 7 = 18, which is divisible by 9, so 4,527 is too.
Convert to a common denominator of 30: 2/3 = 20/30, 3/5 = 18/30 and 7/10 = 21/30. Comparing numerators gives 18 < 20 < 21, so the ascending order is 3/5, 2/3, 7/10.
Write the number as 7k + 3. Doubling gives 14k + 6, and 14k is a multiple of 7, so the remainder is 6. Equivalently, 2 × 3 = 6, which is less than 7 and needs no further reduction.
The least common denominator of 4, 6 and 3 is 12. Converting: 9/12 + 10/12 − 4/12 = 15/12, which reduces to 5/4. Checking with decimals, 0.75 + 0.833 − 0.333 ≈ 1.25 agrees.
For any two positive integers, HCF × LCM equals the product of the numbers. So 6 × 72 = 432 is the product, and the other number is 432 ÷ 18 = 24. Checking, HCF(18, 24) = 6 and LCM(18, 24) = 72, as required.
For a single repeating digit, the fraction is the digit over 9: 0.444... = 4/9. This is already in lowest terms since 4 and 9 share no common factor. Checking, 4 ÷ 9 = 0.444..., as required.
The digit sum of 3,428 is 3 + 4 + 2 + 8 = 17. The next multiple of 9 above 17 is 18, so we need 1 more. Adding 1 gives 3,429 with digit sum 18, which is divisible by 9.
Test each condition. For 7: 7 ÷ 5 leaves 2, and 7 ÷ 3 leaves 1, so both hold. The general solution is 7 + 15k, since 15 is the LCM of 5 and 3.
Divide 5 by 16: 16 goes into 50 three times (48), remainder 2; into 20 once (16), remainder 4; into 40 twice (32), remainder 8; into 80 five times (80), remainder 0. The quotient digits are 0.3125, a terminating decimal.
Write 72 = 2³ × 3². The number of factors equals the product of one more than each exponent: (3 + 1)(2 + 1) = 4 × 3 = 12. Listing them confirms: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
Convert to fractions: 1.2 = 6/5 and 0.25 = 1/4. Multiplying gives 6/20 = 3/10. Checking in decimals, 1.2 × 0.25 = 0.3, and 3/10 = 0.3, which agrees.
For a valid pair, HCF × LCM must equal the product of the numbers, here 15 × 180 = 2700. Also each number must be a multiple of the HCF. For 45 and 60: product 2700, HCF 15, LCM 180, so all conditions hold.
Let x = 0.121212... Then 100x = 12.121212..., and subtracting gives 99x = 12, so x = 12/99. Dividing numerator and denominator by 3 gives 4/33.
Write the numbers as 7a and 7b with a + b = 12 and a, b coprime. The coprime pairs summing to 12 are (1,11), (5,7) and the reverses, giving three unordered pairs; but counting ordered pairs gives six. Careful: the coprime pairs are (1,11), (5,7) and their reverses, six ordered pairs, of which three are unordered. The intended count of unordered pairs is 3, so this option is not correct; the distinct unordered pairs are (7,77), (35,49) and (49,35) which is the same pair, giving 3.
Numbers of the form 9k + 4 are 4, 13, 22, 31, 40, 49, ... Testing each for remainder 5 on division by 11: 49 ÷ 11 leaves 5, since 44 + 5 = 49. Hence 49 is the least solution, and all solutions are 49 + 99k.
Divide 2024 by 12: 12 × 168 = 2016, leaving remainder 8. The next multiple of 12 is 2028, so the amount to add is 12 − 8 = 4. Checking, 2028 ÷ 12 = 169 exactly.
The midpoint of two fractions is their average. Compute 2/5 + 3/7 = 14/35 + 15/35 = 29/35, then halve it to get 29/70. Since 29/70 ≈ 0.414 lies between 0.4 and 0.4286, the answer is consistent.
The product of the numbers is 12 × 144 = 1728, and both must be multiples of 12. Writing them as 12a and 12b with ab = 12 and a, b coprime gives (a,b) = (1,12) or (3,4). These yield pairs (12,144) and (36,48), but only (36,48) has LCM 144 and distinct numbers with HCF 12; sum 84. Rechecking (12,144) gives sum 156. The valid distinct pair with LCM 144 and HCF 12 is (36,48), sum 84, so no option matches and this item needs review.
A number divisible by both 4 and 6 is divisible by their LCM, which is 12. The two-digit multiples of 12 are 12, 24, 36, 48, 60, 72, 84 and 96, which is eight numbers. Since no option equals eight, this item requires review; the count is 8.
Dividing by a decimal is the same as multiplying both numbers by the same power of ten. Multiplying both by 100 gives 36 ÷ 9 = 4, so the quotient is 4.
The next simultaneous ring occurs after the LCM of 18 and 24 minutes. Since 18 = 2 × 3² and 24 = 2³ × 3, the LCM is 2³ × 3² = 72 minutes, which is 1 hour 12 minutes after 9:00 a.m.
Convert to a common denominator of 48: 5/8 = 30/48 and 2/3 = 32/48, so the required fraction lies between them. Now 11/16 = 33/48, which is above 32/48, so it exceeds 2/3; checking each option against both bounds is needed. Comparing with decimals, 5/8 = 0.625 and 2/3 ≈ 0.667; only 13/20 = 0.65 falls strictly between them, while 11/16 = 0.6875 does not.
Numbers leaving remainder 3 on division by 5 are 3, 8, 13, 18, 23, ... Testing each against division by 7, we find 18 = 2 × 7 + 4, so 18 satisfies both conditions. Since the LCM of 5 and 7 is 35, all solutions are 18 + 35k, making 18 the least positive one.
Evaluate the prime factorization: 2^3 = 8, 3^2 = 9, and 8 × 9 = 72, then 72 × 5 = 360. The smallest prime appearing in the factorization is 2, since 2 is the least prime divisor of any even number. Thus the number is 360 and its smallest prime factor is 2.
Let cost be 100. Marked price is 125 after a 25% markup, and a 20% discount reduces it by 25, giving a selling price of 100. Since selling price equals cost, there is neither profit nor loss.
To convert a fraction to a percentage, multiply it by 100. Here 3/8 × 100 = 300/8 = 37.5. So the fraction equals 37.5%.
Let the original price be 100. After a 20% increase it becomes 120, and a 20% decrease of 120 removes 24, leaving 96. The final price is 96, which is 4% below the original 100.
When there is a 20% gain, the selling price equals 120% of cost. So cost = 480 × 100/120 = 400. Checking: 20% of 400 is 80, and 400 + 80 = 480.
The discount is 15% of the marked price, that is 0.15 × 800 = 120. The customer therefore pays 680.
Let B's salary be 100, so A's is 125. The difference is 25, and as a fraction of A's salary this is 25/125 = 0.2, i.e. 20%. The base changes because the comparison is now against A.
Profit is 300 − 250 = 50. Profit percentage is always calculated on cost price, so 50/250 × 100 = 20%.
The gap between the two scores is 15% of the maximum, and this equals 30 + 30 = 60 marks. Hence maximum marks = 60 × 100/15 = 400. The pass mark is then 45% of 400 plus 30, i.e. 210.
Successive increases multiply the original by 1.10 × 1.20 = 1.32. Dividing 26400 by 1.32 gives 20000. Checking: 20000 → 22000 → 26400.
After a 10% discount the selling price is 0.9 × 600 = 540. Since this represents a 20% profit, cost = 540 × 100/120 = 450.
A 10% loss means 720 is 90% of cost, so cost = 720 × 100/90 = 800. To gain 10% he must sell at 110% of 800, which is 880.
Let the marked price be 100. After 20% off it is 80, and 10% off 80 removes 8 more, leaving 72. The total reduction is 28 on 100, so the equivalent single discount is 28%.
Take the number as 100. A 20% increase gives 120, and a 10% decrease of 120 removes 12, leaving 108. The net change is therefore an 8% increase.
Let the cost of one article be 1, so 20 articles cost 20, which is the selling price of 16 articles. Each article then sells for 20/16 = 1.25, a profit of 0.25 on a cost of 1, i.e. 25%.
Percentage scored equals marks obtained divided by maximum marks, multiplied by 100. Here 480/600 × 100 = 80.
Let income be 100, so expenditure is 70 and savings 30. New income is 120 and new expenditure is 77, giving savings of 43. The increase is 13 on 30, i.e. 13/30 × 100 ≈ 43.33%.
The cost after the trade discount is 75% of 400, i.e. 300. Selling at the list price of 400 gives a profit of 100 on a cost of 300, which is 33.33%.
Let the original price be 100 and consumption 100, so expenditure is 10000. At the new price of 125, consumption must be 10000/125 = 80. The reduction is 20 on 100, i.e. 20%.
For every 1000 g the customer pays for, the dealer gives only 800 g, so he saves 200 g of goods. Gain percentage = 200/800 × 100 = 25%, because the gain is measured against the goods actually given.
The marked price is 125% of cost, so cost = 1250 × 100/125 = 1000. The successive discounts give a selling price of 1250 × 0.9 × 0.96 = 1080, which is indeed an 8% gain on 1000, confirming the answer.