Questions & explanations
1. For consecutive integer problems, remember that among any two consecutive integers, one is even and one is odd, so their product is even. Among any three consecutive integers, exactly one is a multiple of 3. Use these properties to check divisibility.
For product of two consecutive integers to be divisible by 6, it must be divisible by 2 and 3. Since one is even, divisible by 2. For divisibility by 3, one of the two must be a multiple of 3. So the pair must include a multiple of 3. That condition is not always true, but the sum is always odd.
2. Trap to Avoid: Misapplying Order of Operations (PEMDAS). Always perform multiplication and division before addition and subtraction, and handle parentheses first. A common trap is to compute 3 + 4 * 2 as 14 instead of 11. Also, be careful with negative signs: -3^2 = -9, not 9.
When simplifying expressions, write each step clearly. Use parentheses to avoid sign errors. For example, ( -3 )^2 = 9, but -3^2 = -9. Also, remember that absolute value acts like parentheses: | -3 - 5 | = | -8 | = 8.
3. Estimate and Eliminate: Use approximation to rule out obviously wrong answers. For example, if a problem involves 37% of 812, estimate as 0.4*800=320. If options are 200, 300, 400, 500, then 300 is close. Then check more precisely if needed.
Estimation is especially useful in data interpretation and percent problems. Round numbers to one or two significant figures. Eliminate options that are too far from your estimate. Then compute exactly if necessary.
4. Back-solve from Options: For multiple-choice problems, especially when the answer is a specific value, substitute each option into the problem to see which one works. This is efficient for equations or word problems where you can test the answer choices.
Start with the middle value (if options are ordered) to narrow down quickly. For example, if options are 10, 20, 30, 40, try 30 first. If too high, try 20, etc. This saves time compared to solving algebraically.
5. Plug in Numbers: When variables are involved, choose simple numbers that satisfy conditions. For example, if a problem says 'a is 20% of b', set b=100, then a=20. This makes calculations concrete and reduces errors.
Plugging numbers turns abstract relationships into concrete arithmetic. Always check that your chosen numbers satisfy all given conditions. Then compute the expression or compare quantities.
6. Strategy: Use 100 as the original number for percent problems. When dealing with percent increase or decrease, setting the original value to 100 simplifies calculations. For example, if a price increases by 20% then decreases by 20%, start with 100 to find the net effect quickly.
Plugging in 100 avoids fractions and makes percent changes easy to compute. After a 20% increase, value becomes 120; after a 20% decrease, it becomes 96, so net 4% decrease.
7. To find numbers with exactly N factors, use prime factorization. If N is prime, the number must be of the form p^(N-1). If N is composite, consider combinations of exponents. For example, exactly 3 factors means exponent 2 (since 3 is prime), so p^2. Exactly 4 factors: p^3 or p*q.
Number of factors = product of (exponent+1). For exactly 3 factors, exponent+1=3 => exponent=2. So p^2. For exactly 4 factors: exponent+1=4 (p^3) or (1+1)(1+1)=4 (p*q).
8. When solving remainder problems, plug in numbers that satisfy the conditions. For example, if n mod 6 = 4 and n mod 8 = 2, list numbers that work for each condition separately and find common ones. Start with the smaller modulus and add that modulus until you find a match.
This technique avoids algebraic manipulation. For n mod 6 = 4: 4,10,16,22,28,... For n mod 8 = 2: 2,10,18,26,... Common: 10, 34, etc. Then test division by 24.
9. When converting a repeating decimal to a fraction, set the decimal equal to x, multiply by a power of 10 to shift the repeating part, subtract to eliminate repetition, then solve for x. For example, 0.7 repeating: x=0.777..., 10x=7.777..., subtract: 9x=7, so x=7/9.
This technique works for any repeating decimal. For mixed repeating decimals (e.g., 0.1666...), multiply by appropriate power to isolate the repeating part.
10. When given GCF and LCM, use the relationship: product of numbers = GCF * LCM. Let the numbers be GCF * x and GCF * y, where x and y are coprime. Then xy = LCM / GCF. List coprime factor pairs of that quotient. This quickly yields possible numbers.
Example: GCF=6, LCM=72 => product=432, xy=12. Coprime pairs: (1,12) and (3,4). Numbers: 6 and 72, or 18 and 24. Sums: 78 or 42. Choose the one in options.
11. Strategy: When given a final value after percent changes, work backwards using inverse operations. For example, if after a 20% increase the price is $120, the original is $120 / 1.2 = $100. If after a 20% decrease the price is $80, the original is $80 / 0.8 = $100.
To reverse a percent increase, divide by (1 + percent/100). To reverse a decrease, divide by (1 - percent/100). This is key for finding original values.
12. Strategy: For successive percent changes, multiply the multipliers. An increase of p% is a multiplier of 1 + p/100; a decrease of q% is 0.9 for 10% decrease. The overall multiplier is the product. For example, a 10% increase then 10% decrease gives 1.1 × 0.9 = 0.99, a 1% decrease overall.
This avoids adding or subtracting percentages incorrectly. Remember that successive percent changes are not additive; you must multiply the factors.