Structural grasp

The pages left in the book, the money left after the lunch box and the seats still free in the classroom are all solved by one subtraction of the form 'whole minus part': 300−120=180 pages, 1000−380=620 yen, 45−28=17 people. Only the bread is 'amount per unit times number of units', a multiplication: 80×6=480 yen, so its type differs. Look only at the operation and the roles of the numbers, not at the subject matter or the size of the figures.

The juice, the car and the eggs are all 'amount per unit times number of units': 120×7=840 yen, 60×3=180 km, 12×5=60 eggs. A speed of 60 km per hour is an amount per unit, namely 60 km for each hour, so the distance problem falls into the same type. Only the coloured paper is 'whole divided by the number of shares', a division: 84÷7=12 sheets, so its form differs from the three multiplications.

The sweets, the water and the money are equal sharing, 'whole divided by number of shares equals the size of one share': 96÷8=12 sweets, 72÷6=12 L, 540÷4=135 yen. The unit of the answer stays the same as the unit of the whole in each case. Only the ribbon is measurement division, 'whole divided by amount per unit equals how many units': 150÷25=6 pieces, and the unit of the answer changes to pieces. The dividing line is whether the divisor is a number of shares or an amount per unit.

The rope, the takoyaki and the cards are measurement division, 'whole divided by amount per unit equals how many units': 24÷3=8 pieces, 36÷4=9 plates, 63÷7=9 people. Each answer is a count. Only the marbles are equal sharing, 'whole divided by number of shares equals the size of one share': 95÷5=19 marbles. Even with a single division in each, the type splits according to whether the divisor is an amount per unit or a number of people.

The example subtracts the amount now in the tank from the full amount, 'whole minus part': 500−180=320 L. The same form is the route problem: 12−5=7 km. The other three are all 'amount per unit times number of units': 150×4=600 yen, 6×7=42 pens, 40×2=80 km. The subjects are scattered across water, distance, cakes, pens and a bus, but in structure only the route problem pairs with the example.

The discount, the attendance and the clay used all have the base amount known and are calculated as 'base amount times proportion', a multiplication: 800×0.3=240 yen, 200×0.7=140 people, 1500×0.2=300 g. Only the number of pupils works backwards from 160, the amount corresponding to the proportion, to find the base amount, a division: 160÷0.4=400 pupils. Whether the base amount is known decides whether you multiply or divide.

The selling price, the school roll and the water at the start are all 'amount corresponding to the proportion divided by the proportion equals the base amount': 600÷0.3=2000 yen, 150÷0.25=600 pupils, 96÷0.8=120 L. The remaining one already has the base amount of 1200 yen and asks for the part that is 45 percent of it, 'base amount times proportion': 1200×0.45=540 yen. The type is settled by whether what is asked for is the base or the part.

The example is 'amount per unit times number of units': 2×9=18 kg. The same form is the handing-out problem: 4×11=44 sheets. The other three are all measurement division, 'whole divided by amount per unit equals how many units': 180÷20=9 pieces, 45÷3=15 benches, 300÷25=12 days. An amount per unit appears in every one of them; what splits the structures is whether you multiply by it or divide the whole by it.

Of distance, time and speed, the walking, the metres-per-minute and the train problems all ask for time by 'distance divided by speed', a division: 12÷4=3 hours, 1200÷80=15 minutes, 450÷15=30 seconds. Only the car problem asks for distance by 'speed times time', a multiplication: 50×2=100 km. In speed problems the type changes according to which of distance, speed and time is being asked for.

The car, the 600 m and the 1200 m problems are all 'distance divided by time equals speed': 150÷3=50 km per hour, 600÷5=120 m per minute, 1200÷8=150 m per minute, and what is asked for is a speed each time. Only the 6 km per hour problem is 'distance divided by speed equals time': 18÷6=3 hours, so what is asked for is a time. One division each, yet what is divided by what differs, so it belongs to a different group.

The average mark, the daily average and the average height are all 'total divided by count equals average', a division: 410÷5=82 marks, 36÷4=9 km, 1288÷8=161 cm. Only the eggs already have the average and use 'average times count equals total', a multiplication: 60×12=720 g. Do not assume that an average problem means division; check what is known and what is asked for.

The sweets, the money and the milk are all 'subtract, then divide': (60−12)÷8=6 sweets, (500−120)÷4=95 yen, (24−4)÷5=4 L. In each you first subtract to get what is left and then share that out. Only the mandarins are 'multiply, then subtract': 1000−90×7=370 yen, where you first multiply to get the price and then subtract. Two-step problems split into groups according to which calculation comes first.

The money, the weights and the rectangle are all of the type where a sum and a difference are given and the larger value is asked for, using '(sum plus difference) divided by 2': (3000+400)÷2=1700 yen, (50+6)÷2=28 kg, (34+4)÷2=19 cm. Only the pen and eraser have both values given outright and need a single subtraction: 280−120=160 yen. The dividing line is whether the individual values have to be recovered from a sum and a difference.

The example is measurement division, 'whole divided by amount per unit equals how many units': 18÷3=6 pieces. The same form is the car problem: 40÷5=8 cars, where the answer is a count of cars. The other three are equal sharing, 'whole divided by number of shares equals the size of one share': 144÷12=12 cookies, 96÷6=16 cm, 3000÷6=500 yen, and the unit of the answer stays the same as the unit of the whole.

The example is 'amount corresponding to the proportion divided by the proportion equals the base amount': 96÷0.3=320 pages. The same form is the sales problem: 280,000÷0.4=700,000 yen. In the other three the base amount is already known and you simply calculate 'base amount times proportion': 2500×0.2=500 yen, 60×0.35=21 pupils, 800×0.15=120 mL. Look at whether the base or the part is being asked for.

The market, the factory and the visitors all end flatly, with 'was held', 'manufactures' and 'was fewer'; each states a fact, something that happened or a present state, as it stands. Only the construction sentence puts だろう (darō, 'probably') at the end and is a conjecture about something ahead that has not yet been confirmed. However scattered the subjects, the groups split purely on whether the ending is flat or conjectural.

The bicycle park, the meetings and the reading time all state the writer's judgement or claim, with 'I think ... should', 'it would probably be better' and 'ought to'. Only the rainfall sentence states a figure flatly as a fact, with none of the writer's thinking in it. Note that 'ought to' counts as opinion even though its ending is a plain declarative, because the word of obligation is there. Look not only at the ending but at whether words of evaluation or obligation appear on the surface.

Three of them put the cause first, 'heavy snow fell', 'no booking could be made', 'the parts did not arrive', and join on the effect with ので (node) or ため (tame), giving cause then effect. Only one states the effect first and flatly, 'the crop was poor', and then explains the cause afterwards with なぜなら〜からだ (nazenara ... kara da, 'that is because'), giving effect then cause, which is the reverse order.

Three of them state the outcome or the action flatly in one sentence first and then add the reason afterwards with からだ (kara da) or ためである (tame de aru), giving effect then cause. Only one puts the cause first, 'enquiries increased', and joins on the effect with ので (node), giving cause then effect. All you need to look at is whether the reason comes before or after.

The example puts the cause first, 'the road was congested', and joins it to the effect with ので (node), giving cause then effect. The only sentence with the same form is the one about the materials, where what precedes ため (tame) is the cause and what follows is the effect. The other three all state the effect flatly first and place the reason afterwards with からだ or ためである, which is the reverse order.

The Japanese high schools, the younger generation and the libraries are general statements that gather many cases together and describe a trend. Only the sentence beginning 'for example' gives one individual case, 'the high school I attend', and is a concrete instance. The groups split according to whether the scope is the whole or a single case.

Three of them take up one particular establishment, 'the bakery in front of the station', 'the bookshop in the next town', 'the coffee shop near my house', and are concrete instances. Only one gathers many shops together as 'shops in the shopping street' and states a trend, which is a general statement. Whether what is pointed at is a single case or the whole is the clue.

Three of them use 場合は (baai wa, 'in the event that'), たら (tara, 'if') and なければ (nakereba, 'if not') to state what happens only when something holds, so they carry a condition. Only one has no word of condition and flatly states something that applies at all times, with no condition attached. All you need to check is whether a conditional word is present.

Three of them carry no word of condition and state flatly something that always holds, so they are unconditional. Only one places a condition, 'if they have a guardian's consent', and states what holds only when that condition is met. Neither the importance of the content nor the length of the sentence matters; the split is purely on the presence of a conditional clause.

The example shows a condition with たら (tara, 'if') and states what happens only when that condition holds. The only sentence with the same form is the one placing the condition 'in the event that the capacity is reached'. The other three have no word of condition and simply state the opening hours, where a form is handed out and where the car park is, with no condition attached.

Three of them state what the writer is setting out to do, with 'I want to', 'I intend to' and 'I am thinking of', so they express intention or wish. Only one ends in ください (kudasai, 'please'); there the one who acts is the reader, and it is a request that asks the other person to do something. Look at who does the acting in the sentence.

Three of them use 'please check', 'please switch off' and 'return it', asking the reader to act, so they are requests or instructions. Only one states the writer's own plan and is a sentence of intention, asking the reader for nothing. The split is mechanical: whether the ending has the form of a request or an order.

Three of them use 'first ... then', 'first ... after that' and 'first ... next' to set out two actions by the same person in time order, so they are sentences of procedure. Only one takes up two different parties, the elder and younger brother, and marks the difference with 'but', making it a contrast; nothing is said about what comes before or after in time.

The range and the prices, Company A and Company B, and last year and this year all set two matters side by side with 'but' or 'while' and mark the difference, so they are contrasts. Only the envelope sentence sets two actions in time order with 'after that' and makes no comparison.

The example joins a merit, 'is light', and a drawback, 'durability is not high', with 'but', setting two sides of one thing against each other, so it is a contrast. The same form is the electronic dictionary sentence, which sets the number of entries against the speed of response. The other three join cause and effect with ので (node) or ため (tame), giving cause then effect, and compare nothing.