Unit 5: Mathematical Reasoning and Aptitude mind map

Find the rule of a number series, a letter series or a mixed series, and spot the wrong term.

Number series

Differences and second differences

In many series, the gaps between terms change by a fixed amount. Find the gaps, then the gaps of the gaps.

• Write the differences between neighbouring terms.
• If the differences are not equal, take their differences again.
• Example: 2, 9, 20, 35, 54, 77 has differences 7, 11, 15, 19, 23.
• These rise by 4, so the next difference is 27 and the next term is 104.

A constant second difference means the terms follow a quadratic pattern. The series 1, 6, 15, 28, 45 has differences 5, 9, 13, 17, so the next difference is 21 and the next term is 66.

Shrinking and growing gaps

Sometimes the gaps are negative or shrink. The same method works.

• Example: 56, 40, 27, 17, 10 has gaps -16, -13, -10, -7.
• The gaps rise by 3, so the next gap is -4.
• The next term is 10 - 4 = 6.
• Check by following the rule backward.

Do not assume the series keeps falling at one speed. Write the gaps and check whether they follow their own pattern.

Multiply and add

Some series multiply the previous term and then add or subtract a fixed number.

• Example: 6, 13, 27, 55, 111 uses 'double and add 1'.
• The next term is 111 x 2 + 1 = 223.
• A series 1, 3, 7, 15, 31 is 2^n - 1, and the next is 63.

If the terms roughly double, test 'times 2 plus something'. A good check is to compare the first two terms and try the same rule on the next pair.

13 = 6 x 2 + 1, 27 = 13 x 2 + 1, and so on.

Squares, cubes and powers

Many series are squares or cubes with a small change.

• n squared minus 1: -1, 0, 3, 8, 15, 24, then 35.
• Even numbers squared minus 1: 3, 15, 35, 63, then 99.
• n cubed plus n: 2, 10, 30, 68, 130, then 222.
• Squares of half-numbers: 0.25, 2.25, 6.25, 12.25, then 20.25.

Learn the squares up to 15 and cubes up to 10. A term that is one below or one above a square is a strong hint.

Series made of two series

When odd and even terms behave differently, split the series into two.

• Take the terms in odd places and the terms in even places separately.
• Squares and cubes can alternate: 4, 27, 16, 125, 36, ?, 64, 729.

Check which place the missing term falls in. Then continue only that series.

Example: 5, 9, 3, 11, 1, 13, -1, ? has odd terms falling by 2 and even terms rising by 2. The missing term is 15.

Another: 2, 4, 6, 12, 10, 36, 14, ?, 18, 324 has odd terms +4 and even terms times 3. The missing term is 108.

Wrong term in a series

One term is wrong, and you must find which one breaks the rule.

• Find the rule from the terms you trust.
• Gaps that should fall by 11 each time reveal the term that breaks the pattern.

Look for the term that does not fit the pattern of all the others. Test the rule on the first few terms, then find where it fails.

A series of primes with one composite: 3, 13, 43, 53, 63, 83 has 63, which is 7 x 9.

A series made by multiplying by successive primes: 6, 18, 90, 630, 6930 should continue to 90,090, not 83,160.

Letter and alphanumeric series

Convert letters to numbers

To solve a letter series, replace each letter with its place in the alphabet.

• A = 1, B = 2, ... Z = 26.
• Treat each letter position as its own number series.
• Example: AA, BD, CI, DP has second letters 1, 4, 9, 16, the squares.
• The next second letter is 25, which is Y, so the next term is EY.

Write the numbers under the letters. Most letter series then become number series that you already know how to solve.

Letter groups and steps

Letter groups often move by a fixed step in each position.

• JAK, KBL, LCM, MDN: each letter moves up one place, so the next is NEO.
• DEF, HIJ, LMN, PQR, TUV: blocks advance by 4, so the next is XYZ.

Look at each position separately. If a position passes Z, wrap around to A and keep counting.

ME, KH, IK, GN: first letters fall by 2, second letters rise by 3, so the next is EQ.

KB, JH, IM, HQ, GT: the second letter gaps are +6, +5, +4, +3, so the next is FV.

Alphanumeric series

In an alphanumeric series, treat the numbers and the letters as separate series.

• Example: 1D3, 3H5, 5L7.
• First numbers 1, 3, 5, so next 7.
• Letters D, H, L go up by 4, so next P.
• Last numbers 3, 5, 7, so next 9. The term is 7P9.

Another series, N7V, K9T, H12R, ?, B21N, had letters falling by 3 and numbers rising by 2, 3, 4, 5. The missing term was E16P.

Find the rule that turns a word into a code, then apply it to a new word.

Letter-value codes

Sum of letter positions

In a sum code, the code number is the sum of the alphabet positions of the letters.

• A = 1, B = 2, ... Z = 26.
• BAT = 2 + 1 + 20 = 23.
• BALL = 2 + 1 + 12 + 12 = 27.
• Always check the rule on a second example.

A code may be the sum with a twist, such as the sum divided by 5. Test the plain sum on each given example first.

Product and average of positions

Some codes multiply the letter positions or take their average.

• HOTEL: sum 60, average 12. BALM: sum 28, average 7.
• For DU, the sum is 25 and the code is 5.

When the code is much smaller than the sum, test the average or a divisor. When it is much larger, test the product.

RED = 18 x 5 x 4 = 360. BLUE = 2 x 12 x 21 x 5 = 2,520.

HPU and JNU both sum to 45 and code to 9, so the code is sum divided by 5.

Position codes and reversals

Some codes write the letter positions after a change, such as adding a number, doubling or reversing.

• GERMANY coded 9, 7, 20, 15, 3, 16, 27: every position plus 2.
• EARTH coded 102364016: each position doubled and written together.
• WHERE coded 5, 18, 5, 8, 23: positions written backwards.
• Check the rule on two words before applying it.

Compare each letter's position with its code. If the same operation fits every letter, you have the rule.

Letter-shift codes

Uniform and alternating shifts

In a shift code, each letter moves forward or backward by a fixed amount.

• SUNDAY to TVOEBZ: each letter moves +1.
• BEAUTIFUL to ADZTSHETK: each letter moves -1.
• STORK to VQRON: shifts +3, -3, +3, -3, +3.
• CREDIT to EPGBKR: shifts +2, -2, +2, -2.

Write each shift under the letter. If the shifts alternate, apply the same alternating pattern to the new word.

Progressive shifts

In a progressive shift, the shift grows by one each time.

• DICE to FLGJ: +2, +3, +4, +5.
• GLAMOUR to HNDQTAY: +1, +2, +3, +4, +5, +6, +7.
• For a longer word, keep adding one more to the shift.
• Wrap around after Z.

If each letter moves by a different amount, list the shifts. They will often be 1, 2, 3 or 2, 3, 4.

Letters expanded and halves swapped

Some codes expand each letter into two, or swap the two halves of a word.

• LIFE becomes KMHJEGDF: each letter is replaced by the letter before and the letter after.
• PLEADING becomes FMHCQMFB: the halves swap, and each half is shifted.
• For WORD, the code is VXNPQSCE.
• Always count the letters in the code first.

If the code is twice as long as the word, each letter gives two letters. If the code has the same length, each letter gives one.

Substitution codes

Word and sentence codes

In substitution codes, words in sentences are coded as other words or numbers. Find which word matches which code by comparing sentences.

• Find a word that appears in two sentences.
• Find the code that appears in both codes.
• That word and code match.
• Remove it and repeat.

Do not match words by their position in the sentence. Match them by what the sentences share. If 'carpet' is in all three sentences and the digit 6 is in all three codes, then carpet = 6.

Letter-to-digit tables

In some codes, each letter stands for one digit. Build a table from the given examples.

• Write every letter and its digit from the examples.
• Check that the same letter always gives the same digit.
• Then code the new word using your table.

Use letters that appear in two words to confirm your table. The table must be consistent across all examples.

Example: CRICKET = 2632479 gives C = 2, R = 6, I = 3, K = 4, E = 7, T = 9.

Averages, ratios, proportions, fractions and percentages, with the steps NTA's own solutions use.

Averages

The basic average formula

An average is the total divided by the number of items. Always work with totals.

• Average = total / number of items.
• Total = average x number of items.
• Example: five numbers with average -20 have total -100.
• If three add to 32, the other two add to -132, and their average is -66.

Convert each average to a total first. Then add or subtract totals, and divide by the new count.

Combined and weighted averages

When two groups are merged, find each group's total and divide by all the items.

• Cadre A: 40 employees, mean 31, total 1,240.
• Cadre B: 60 employees, mean 28, total 1,680.
• Combined: 2,920 / 100 = 29.2.

Do not average the two averages. The bigger group counts for more. Take 100 people when percentages are given.

Percentage shares work the same way: 40% workers at Rs 39,000 and 60% executives at Rs 42,000 give Rs 40,800.

A person joins, leaves or is replaced

When a person is added or replaced, the total changes by that person's value.

• Average age problems: add each person's years when the time moves.
• A baby's age is the new total minus the old total.

Write one equation for the total before and one for the total after. The difference is the added or removed value.

Missing score: total of 35 scores is 630. The first 17 add to 238 and the last 17 add to 340. The 18th score is 630 - 578 = 52.

A batsman's 87 in the 17th innings raised his average by 3. Then 16x + 87 = 17(x + 3), so x = 36 and the new average is 39.

Average speed

When the distances are equal, average speed is not the simple average of the speeds.

• Average speed = total distance / total time.
• For equal distances at speeds a and b: 2ab / (a + b).
• At 30 km/h and 45 km/h: 2 x 30 x 45 / 75 = 36 km/h.
• At 40 and 60 km/h for 360 km each way: 720 / 15 = 48 km/h.

A car spends more time at the slower speed, so the average is closer to the slower speed. The harmonic mean formula handles this.

Averages of number sets

Some averages have shortcuts for special sets of numbers.

• Consecutive odd or even numbers: the average is the middle number.
• Five consecutive odd numbers with average 95 are 91, 93, 95, 97, 99.
• With sum 225 the middle is 45, and the numbers are 41, 43, 45, 47, 49.

For grouped data, multiply each mid-value by its frequency, add, and divide by the total frequency. That gave a mean of 59.4 kg for 60 students.

Grouped data: use the middle value of each class. Weights 50-54, 54-58, 58-62, 62-66 use 52, 56, 60, 64.

Ratio and proportion

Dividing in a ratio

To divide a quantity in a ratio, add the parts, then find one part.

• Ratio 3 : 5 : 7 has 15 parts.
• If the sum is 975, one part is 65.
• The numbers are 195, 325 and 455.
• For coins, find the value of one set first.

Coins of 5 paise, 10 paise and 25 paise in the ratio 3 : 2 : 1 make 60 paise per set. Rs 60 is 6,000 paise, so there are 100 sets and 300 five-paise coins.

When a number is added to a ratio

If the same number is added to both terms, write the numbers as multiples and solve.

• Two numbers in the ratio 3 : 7 are 3k and 7k.
• After adding 8, (3k + 8) / (7k + 8) = 5/9.
• Cross-multiply: 27k + 72 = 35k + 40, so k = 4.
• The numbers are 12 and 28.

Always check by putting the numbers back. 20/36 reduces to 5/9, so the answer fits.

Fourth and third proportional

A proportion links four quantities. There are two special formulas.

• Fourth proportional to a, b, c is bc / a.
• Third proportional to a and b is b squared / a.
• Fourth proportional to 4, 9, 12 is 27.
• Third proportional to 16 and 36 is 81.

In a chain of proportions, two inversions cancel. If a is inversely proportional to b, and b is inversely proportional to c, then a is directly proportional to c.

Ratios in word problems

Put numbers in the ratio to get a simple case.

• If x : y = 7 : 9, put x = 7 and y = 9.
• Then 3x - 5y : 4x + y = -24 : 37.
• Class with 4 boys to 3 girls: take 4 and 3 students.
• On a map, 1 inch for 250 miles makes 4.5 inches stand for 1,125 miles.

Choosing a small whole-number case saves time. Percentages of a mixed class become easy when you pick 4 boys and 3 girls.

Fractions

Ordering fractions

To order fractions, convert each to a decimal.

• 12/19 = 0.632, 7/12 = 0.583, 11/17 = 0.647, 17/28 = 0.607.
• Decreasing order: 11/17, 12/19, 17/28, 7/12.
• If denominators are the same, the bigger numerator is bigger.
• A fraction is reduced by dividing numerator and denominator by a common factor.

This is one of the most common question types. Divide, keep three decimals, and compare. Questions often give close values, so keep enough digits.

Working back from a fraction

When a part is given away, work backwards from what is left.

• Rohan gave away three-fifths and has Rs 2,000 left, which is two-fifths.
• So the whole is Rs 5,000, and he gave Rs 3,000.
• The dog's share is T/7, so the fortune T is Rs 1,40,000.

Track what remains after each step. The last remaining fraction equals the last known amount.

In a fortune question, half goes to the wife, 1/7 of the rest to the son and 2/3 of the rest to the daughter. The dog gets Rs 20,000.

Percentages

Successive percentage changes

Percentage changes multiply. They do not add.

• A 5% rise then a 5% fall: 1.05 x 0.95 = 0.9975.
• A 20% rise then a 25% fall: 1.20 x 0.75 = 0.90, a 10% fall.
• Net change = (factor - 1) x 100.
• Use factors for any number of changes.

Turn each change into a factor, such as 1.05 for +5% and 0.95 for -5%. Multiply all the factors. Do not add the percentages.

Successive discounts

Successive discounts also multiply. Find what remains, then subtract from 100.

• Discounts of 15%, 20% and 40% make 59.2%.
• Discounts of 10% and 15% make 23.5%.

A single equivalent discount is always less than the sum of the discounts. For example, 20%, 10% and 5% make 31.6%, not 35%.

Three discounts of 30%: 0.7 x 0.7 x 0.7 = 0.343, so the single discount is 65.7%.

Discounts of 10%, 20% and 30%: 0.9 x 0.8 x 0.7 = 0.504, so the single discount is 49.6%.

Percentages in area and other changes

When two quantities change by percentages, their product changes by the product of the factors.

• Length +0.8% and width -0.5% change the area by 1.008 x 0.995 = 1.00296, about +0.3%.
• 43% - 28% = 15% of a number. If that is 75, the number is 500.

Always find the base. A percentage is useless until you know what it is a percentage of.

A price rise of 12% and a discount of 20% leave 1.12 x 0.8 = 0.896, a 10.4% fall.

If 60% of (a - b) = 40% of (a + b), then b is 20% of a.

Percentage word problems

Treat percentages as parts of a whole and write one equation.

• 65% of N is 21 less than 4/5 of N: N = 140.
• Convert units before finding a percentage: 72 seconds is 4% of 30 minutes.

Let the total be N. Express every group as a fraction of N. The words 'in all' or 'did not' tell you which group to add.

Commission: 10% on the first Rs 10,000 and 5% above that. S - 1,000 - 0.05(S - 10,000) = 75,500 gives S = 80,000.

60% female and 40% male. 40% of females and 70% of males opted for a course. 60 did not opt, so N = 125.

Cost price, selling price, marked price, discount, simple and compound interest.

Profit, loss and discount

Cost price, selling price and profit

Profit and loss are always worked on the cost price.

• Profit percent = profit / cost price x 100.
• Selling price = cost price x (1 + profit%).

If profit at one price equals loss at another, the cost price is exactly halfway. Here (2,220 + 1,580) / 2 = Rs 1,900. A 25% profit then gives Rs 2,375.

Rs 22,000 at a 10% profit means a cost of 20,000. At an 18% profit the price is Rs 23,600.

Loss and gain on the same cost: the cost is halfway between the two prices if the percentages are equal.

Marked price and discount

The marked price is the printed price. A discount is cut from it.

• Selling price = marked price x (1 - discount%).
• To gain 8% after a 10% discount, mark 20% above cost.

Take the cost price as Rs 100. Find the marked price, then the selling price, and compare.

Marked 40% above cost, then a 20% discount: 140 x 0.8 = 112, a profit of 12%.

A 25% discount gives Rs 36, so MP = 48. A 20% discount then gives Rs 38.4.

Profit on mixtures and other problems

For mixtures and unusual cases, find the cost of everything, then compare with the sale.

• 70 pencils for Rs 90 at 25% loss: cost Rs 120.
• Two shopkeepers with 20% and 15% discounts give 32%. Discounts of 10% and 25% give 32.5%.

Write the cost price as C. Express both selling prices in terms of C and subtract.

Oils at Rs 100, 40 and 60 in the ratio 2 : 4 : 3 cost Rs 540 for 9 litres, Rs 60 per litre. Selling at Rs 66 gives a 10% profit.

If a loss of 25% becomes a gain of 10% by Rs 175 more, then 0.35C = 175 and C = Rs 500.

Simple and compound interest

Simple interest formula

Simple interest = P x R x T / 100.

• P is the principal, R the rate per year and T the time in years.
• Rs 2,000 yields Rs 180 in 9 months. T = 0.75, and R = 12%.
• Convert months to years before using the formula.

If the interest and time are given, find R. If the interest, rate and time are given, find P.

Interest is proportional to time, so the ratio of interest for 9 and 12 years is 3 : 4.

Finding the sum and rate from two amounts

When a sum grows to two amounts at two times, the difference gives the interest for the gap.

• Rs 1,008 after 2 years and Rs 1,164 after 3.5 years.
• Interest for 1.5 years is Rs 156, so Rs 104 per year.
• Interest for 2 years is Rs 208, so the principal is Rs 800.
• The rate is 104 / 800 = 13%.

Simple interest grows by the same amount each year. Find that yearly amount first, then work back to the principal.

Splitting a sum at two rates

When a sum is split into two parts at two rates, let one part be P and the other the rest.

• Rs 16,800 split at 6% and 8%. After 2 years, the total is Rs 19,000.
• Part at 6% grows by 1.12. Part at 8% grows by 1.16.
• 1.12P + 1.16(16,800 - P) = 19,000 gives P = 12,200.
• The other part is Rs 4,600.

Write the amount factor for each part. Set the total equal to what is given and solve for P.

Compound interest

Compound interest is on the amount of the previous period. Amount = P x (1 + r)^n.

• Rs 10,000 at 8% half-yearly for 1.5 years: rate 4% per half-year, 3 periods.
• Amount = 10,000 x 1.04^3 = Rs 11,248.64.
• If the interest is paid half-yearly, halve the rate and double the periods.
• Effective annual rate for 10% half-yearly is 10.25%.

For quarterly compounding, divide the rate by 4 and multiply the years by 4. The effective rate is higher than the nominal rate.

Compound interest from two amounts

The ratio of two successive amounts is the growth factor for one year.

• To double in 4 years: (1 + r)^4 = 2, so r is about 18.9%.
• To double in 14 years, the rule of 72 gives about 5%.

The rule of 72 says that the doubling time is about 72 divided by the rate. It is a quick check, not an exact answer.

Rs 10,816 after 3 years and Rs 11,248.64 after 4 years: 11,248.64 / 10,816 = 1.04. The rate is 4%.

Rs 2,400 after one year and Rs 3,000 after two: the factor is 1.25 and the principal is Rs 1,920.

Difference of compound and simple interest

Over two years, the difference between compound and simple interest is P x (r/100) squared.

• At 10%, the difference is P x 0.01.
• If the difference is Rs 5,431, then P = Rs 5,43,100.
• Simple interest of Rs 800 at 5% for 4 years means P = Rs 4,000.
• Compound interest on it is 4,000 x (1.05^4 - 1) = Rs 862.

For two years, the extra comes only from interest earned on the first year's interest. That is why the formula uses the rate squared.

Work done together, pipes, chain rule, speed and distance, trains and boats.

Time and work

Work rates add

If a person finishes a job in n days, his work per day is 1/n. Rates of people working together add.

• Together they do 21/30 of the job per day.
• So they finish in 30/21 = 1 3/7 days.

Always find the rate for the whole job first. Then add the rates and take the reciprocal.

A, B and C take 3, 5 and 6 days. Their rates are 1/3, 1/5 and 1/6.

X does 1/3 of a job in 5 days, so the whole job takes 15 days. Y does 2/5 in 10 days, so 25 days. Together 75/8 = 9 3/8 days.

Solving for an unknown joint time

When two people together take n days, and alone they take n + 3 and n + 12, set up an equation.

• 1/(n + 3) + 1/(n + 12) = 1/n.
• Cross-multiply: n(2n + 15) = (n + 3)(n + 12).
• This gives n squared = 36, so n = 6.
• Check: 1/9 + 1/18 = 1/6.

The n terms cancel and a simple square remains. Always check your answer in the original equation.

Pipes and tanks

Pipes that fill a tank are like workers. Pipes that empty it have negative rates.

• Pipe A fills in 3 hours, pipe B in 5 hours.
• A opens at 7 am, B at 8 am. By 8 am A has filled 1/3.
• Together they fill 8/15 per hour. The remaining 2/3 takes 1.25 hours.
• The tank is full at 9:15 am.

Do the first part alone. Then, for the rest, add the rates of the pipes that are open together.

Work shares and alternating work

When money is shared for work, share it by work done.

• X did 4/6 = 2/3 of the work, worth Rs 2,000.
• The assistant's share is Rs 1,000.
• For alternating taps, find the work done in one full cycle.

The assistant's share is the part of the work he did, not the time he worked. Work out each person's fraction first.

X can do a Rs 3,000 job in 6 days. With an assistant, it takes 4 days.

Chain rule

In a chain rule problem, the total work equals men x hours x days.

• 5 pumps x 18 hours x 12 days = 1,080 pump-hours.
• With 6 pumps for 9 days, the hours per day are 1,080 / 54 = 20.
• 15 men x 8 hours x 18 days = 2,160 man-hours.
• For 6 days at 12 hours a day, you need 30 men.

Find the total work in man-hours, pump-hours or person-days. Then divide by the new conditions.

Consumption and rate problems

For food or fuel, find the use per person per day.

• 90 persons eat 810 kg in 27 days. One person eats 1/3 kg a day.
• 60 persons eating 300 kg need 300 / 20 = 15 days.
• 40 people eat 320 kg in 50 days. One person-day uses 0.16 kg.
• So 80 kg for 30 people lasts 16 2/3 days.

Treat a person-day as one unit. Find how much one unit consumes, then scale.

Speed, distance and trains

Speed, time and distance basics

Distance = speed x time. Convert units before you calculate.

• km/h to m/s: multiply by 5/18.
• m/s to km/h: multiply by 18/5.
• 54 km/h = 15 m/s. 90 km/h = 25 m/s. 108 km/h = 30 m/s.
• A man walks 750 m in 2.5 minutes, which is 18 km/h.

Always write the units. A common error is mixing minutes with hours or metres with kilometres.

Late and early arrival

When two speeds give a late and an early arrival, the time difference equals the sum.

• At 4 km/h he is 30 minutes late. At 6 km/h he is 5 minutes early.
• The difference in time is 35 minutes = 7/12 hour.
• d/4 - d/6 = 7/12 gives d = 7 km.
• At 3 and 5 km/h with 28 minutes late and early, d = 7 km.

Add the late time and the early time to get the difference between the two journeys. Then set d/slow - d/fast equal to it.

Meeting and overtaking

Two trains meeting have a combined speed. Overtaking uses the difference of speeds.

• Two trains approaching: add the speeds.
• Trains in the same direction: subtract the speeds.

If one train starts earlier, take its head start away from the distance first. Then use the combined speed.

Train 1 leaves at 7:00 at 80 km/h. Train 2 leaves 30 minutes later at 90 km/h. 380 km apart.

The gap left at 7:30 is 340 km. The closing speed is 170 km/h, so they meet at 9:30.

Trains crossing platforms and poles

A train crossing a platform covers its own length plus the platform's length.

• 200 m train, 1,000 m platform, 20 m/s: 1,200/20 = 60 seconds.
• Crossing a pole or man: only the train's length.

Count gaps, not poles. n poles make n - 1 gaps.

A train at 63 km/h takes 20 s to cross a platform. It covers 350 m, so the platform is 250 m if the train is 100 m.

13 poles have 12 gaps. At 125 m each in 2 minutes, the speed is 45 km/h.

Boats and streams

Downstream speed is boat plus stream. Upstream speed is boat minus stream.

• Boat 5 km/h, stream 1 km/h: downstream 6, upstream 4.
• For a 24 km trip each way: 4 + 6 = 10 hours.
• Times in the ratio 3 : 5 mean speeds in the ratio 5 : 3.

Time is inversely proportional to speed for the same distance. If the downstream time is smaller, the downstream speed is larger.

Downstream in 5 h and upstream in 6 h with stream 4 km/h: 5(b + 4) = 6(b - 4), so b = 44.

Round trips and speed puzzles

For round trips and speed puzzles, set up one equation for time.

• A car goes at 40 km/h and returns at 60 km/h. Average speed is 48 km/h.
• Cyclists at 15 and 16 km/h, 16 minutes apart: d = 64 km.

Write the time for each leg. Then use the total or the difference, whichever the question gives.

If you walk one way and ride back in 40 minutes, and walking both ways takes 60, riding both ways takes 20 minutes.

A person in a car at 36 km/h covers half the journey in 4/5 of the time. The other half needs 90 km/h.

Identities, equations, number properties, areas and volumes, mixtures and ages.

Algebra

Useful identities

A few identities solve most algebra questions quickly.

• (a - b) squared = a squared + b squared - 2ab.
• x cubed + 1/x cubed = (x + 1/x) cubed - 3(x + 1/x).
• a cubed + b cubed = (a + b)(a squared - ab + b squared).
• a cubed - b cubed = (a - b)(a squared + ab + b squared).

If x + 1/x = -4, the sum of cubes is (-4) cubed - 3(-4) = -52. If ab = 21 and a squared + b squared = 58, then (a - b) squared = 16.

Simplifying cube expressions

When an expression has cubes in the top and bottom, look for a common factor.

• With 4 times, the answer is 1/64.
• Sum of cubes over a squared minus ab plus b squared is a + b.

Check that the middle term matches the identity. Then the whole fraction becomes a simple sum or difference.

In the first fraction, the bottom numbers are 3 times the top numbers. So the bottom is 27 times the top, and the answer is 1/27.

(0.96 cubed - 0.1 cubed) / (0.96 squared + 0.096 + 0.1 squared) = 0.96 - 0.1 = 0.86.

Equations from words

Turn a word problem into one or two equations.

• Birds on two trees: A = 10 and B = 14.
• Hits and misses: h = 70 and m = 30.

Add and subtract two equations to make the numbers simple. Always put the answer back to check.

Cows and ducks: 60 heads and 180 legs. If all were ducks, 120 legs. Extra 60 legs, each cow adds 2, so 30 cows and 30 ducks.

Potatoes and tomatoes: 3p + 4t = 100 and 4p + 3t = 110. Adding gives p + t = 30. Subtracting gives p - t = 10. So t = Rs 10.

Ages

In an age problem, write each person's age with one letter and change it by the years that pass.

• Father 3 times son, double after 15 years: sum is 60.
• Everyone's age rises by the same number of years.

Use the younger person as the unknown. The older person's age is then a multiple or a sum.

Father is twice the boy. After 6 years the ratio is 11 : 6. (2x + 6) / (x + 6) = 11/6 gives x = 30.

A man of 50 with a son aged 2/5 of that (20). After x years (50 + x) / (20 + x) = 5/3, so x = 25.

Indices and exponents

Convert every number to the same base. Then compare or equate the powers.

• 1024 to the power -1/5 = 2^-2 = 1/4.
• (8/125) to the power -1/3 = 5/2.

Learn the powers of 2 and 3. A square root is the power 1/2 and a cube root is the power 1/3.

16 to the power 0.32 times 4 to the power 0.36 = 2^1.28 x 2^0.72 = 2^2 = 4.

If 2^(x squared) x 8^2 / 8^(2x) = 1/8, write all as powers of 2. x squared - 6x + 9 = 0, so x = 3.

Numbers and series

Divisibility, primes and remainders

Use divisibility rules to find missing digits.

• Remainder 5 on dividing by 8 means remainder 1 on dividing by 4.
• A prime is divisible only by 1 and itself. 63 is not prime.

Remainders can be reduced to a smaller divisor if that divisor divides the larger one. This simplifies many problems.

Divisible by 90 means divisible by 10 and 9. For 345xy, y = 0 and 3 + 4 + 5 + x = 18 gives x = 6.

A number that is both a square and a cube is a sixth power. The smallest above 1 is 64.

Sums of numbers

Sums of the first natural, odd and even numbers have simple formulas.

• Sum of first n natural numbers: n(n + 1)/2.
• Sum of first n even numbers: n(n + 1). Even numbers to 80 add to 1,640.
• Arithmetic series: n/2 x (first + last). 50 terms from 3 to 199 add to 5,050.

The nth term of an arithmetic series is a + (n - 1)d. In a 100-term series from 7 to 205, d = 2 and the 71st term is 147.

Sum of first n odd numbers: n squared. The odd numbers up to 30 add to 225.

Telescoping sums and doubling

Telescoping sums collapse into two terms.

• 1/(5x6) + 1/(6x7) + ... + 1/(24x25) = 1/5 - 1/25 = 4/25.
• 1/n - 1/(n + 1) = 1/(n(n + 1)).
• Bacteria that double every 2 minutes take 20 minutes to go from 1 to 1,024.

For growth that doubles, work backward. One step before full is half full, and two steps before is a quarter.

Doubling problems: if a container fills in 30 minutes and doubles each minute, it is a quarter full at 28 minutes.

Mensuration

Areas and perimeters

Learn the area and perimeter formulas and convert all units to the same unit first.

• Rectangle: length x width. Square: side squared; diagonal d gives d squared / 2.
• Circle: pi r squared and perimeter 2 pi r.
• Equilateral triangle: root 3 / 4 x side squared.
• A circle's perimeter to a square's perimeter, when radius equals side: 11 : 7.

A room of 3 m x 3 m is 300 cm x 300 cm. Slabs of 20 cm x 30 cm number 90,000 / 600 = 150.

Comparing areas and volumes

To compare shapes, work out each area in the same unit.

• Curved surface of a cylinder: 2 pi r h.
• Sphere: 4 pi r squared. Rhombus: half the product of diagonals.
• Put the values in order, then match them to the letters.

Convert everything to the same unit, then use pi = 3.14 or 22/7. Decide the order only after you have all values.

Circle of diameter 10 cm: 78.5. Rectangle 6 x 4: 24. Square side 5: 25. Equilateral triangle side 5: 10.8.

Mixtures and alligation

Cost of a mixture

The cost per unit of a mixture is the total cost divided by the total quantity.

• 10 kg at Rs 50 and 15 kg at Rs 20: Rs 32 per kg.
• For a target price, solve for x: x = 45 kg.

Take a convenient total, such as 5 litres for a ratio of 3 : 2. Find the cost, then divide.

Oil A and B in the ratio 3 : 2 at Rs 60 and Rs 120. 3 x 60 + 2 x 120 = 420 for 5 litres, Rs 84 per litre.

Alligation

Alligation finds the ratio in which two items must be mixed to get a target value.

• Mean price 64, cheaper 60, dearer 78.
• The ratio is 14 : 4 = 7 : 2.
• The mean is first found from the selling price and the profit.

The distance of each from the mean gives the part of the other. Draw the cross to avoid mixing up the order.

The parts are (78 - 64) = 14 for the cheaper and (64 - 60) = 4 for the dearer.

Changing a mixture

When you add or remove parts, track milk and water separately.

• A mixture sold removes both milk and water in the same ratio.
• Adding free water keeps the total cost the same.

The thing that does not change tells you what to hold fixed. In the first case, milk stays 12 litres.

27 litres in the ratio 5 : 4 holds 15 litres of water and 12 of milk. To make water to milk 3 : 1, water must be 36. Add 21 litres.

18 litres of milk at Rs 70 is worth Rs 1,260. To bring the value to Rs 60 per litre, the volume must be 21. Add 3 litres of water.

Direction sense, blood relations, odd one out, data sufficiency and true or false statements.

Direction sense and relations

Direction sense

In a direction question, draw the path with north at the top and track east-west and north-south separately.

• A man walks 1 km east, 5 km south, 2 km east and 9 km north.
• East total 3 km. North-south net 4 km north.
• Distance from start = square root of (9 + 16) = 5 km.

Write the facing direction after each turn. Then add up the movement in each direction, and use Pythagoras for the distance.

A turn to the right of facing south is west; a turn to the left of facing south is east.

Rotated directions

If every direction is rotated by the same angle, rotate north by that angle too.

• South-east becomes east, and north-west becomes west.
• Each direction turns 45 degrees anticlockwise.
• So north becomes north-west.
• A full circle has eight directions, 45 degrees apart.

Find how many degrees one direction moved. Apply the same turn to the direction you are asked about.

Blood relations

In a blood relation question, translate the clues into a family tree.

• 'The only son of my mother' means the speaker himself.
• A man's granddaughter is the only daughter of Abhijit's brother, so the man is Abhijit's father.
• Ram, an only child, points to the husband of his mother's grandchild: his son-in-law.

Work out whether each person is male or female, and what generation they are in. The answer is the relation between the two people asked.

'His wife is the only daughter of my father' means the speaker is a woman and the man is her husband.

Odd one out and sufficiency

Odd one out

In odd-one-out questions, find the property that three words share.

• Paper, wool and jute are natural. Plastic is synthetic.
• Tall, huge and thin describe size. Sharp does not.
• Flourish, prosper and thrive mean to do well. Excite does not.
• Unique, rare and exceptional mean unusual. Beautiful does not.

Decide what the three have in common. The odd one is the one that does not fit that idea.

Analogy and patterns

An analogy compares two pairs. Find the relation in the first pair and apply it to the second.

• Cup is to coffee as bowl is to soup: a container and what it holds.
• Unscramble EIWNTR, UMRSME, PIGRSN, LCUOD: winter, summer, spring and cloud. Cloud is not a season.
• Number patterns can be about primes, squares or other properties.

Put the relation into one sentence, such as 'a container and what it holds', and test every option against it.

The set (31, 19, 41) is made of three primes. The similar set is (29, 17, 37).

Data sufficiency

A data sufficiency question asks which statements are needed to answer the question.

• Work out what quantity is asked.
• Check whether the statements give enough to find it.
• Sometimes either of two statements can replace each other.

Do not solve for the final number if you do not need it. Only decide whether the information is enough.

To find the speed of a train: a pole crossing in 10 seconds and a 300 m platform in 30 seconds give 15 m/s.

True or false statements

In statement questions, check each statement separately.

• 'x% of y' is xy/100, so x% of y is not y% of 100x.
• 20 students with 5th from the top is 16th from the bottom.
• 108 km/h is 30 m/s, so a train takes 15 seconds to cross 450 m.
• Do not judge by looks. Work each statement out.

Take each statement on its own and compute. Then look for the option that matches the true ones.