# Word Problems Explanations

On this page you can find the explanations of the easy, medium and hard word problem tests

## Explanations Word Problems Easy

**1.**Leonardo arrives 12 minutes too late for the 14:12 train from Pisa to Milan. This train runs every hour at 12 minutes past the hour. How long must he wait for the next train?

**Solution:**

The train arrives every hour. Leonardo is 12 min late.

So 1 hour – 12 min = 60 min – 12 min = 48 min

**2.** Leonardo buys 6 abacuses for € 52 each. He sells all 6 for a total of € 372. How much does Leonardo make per abacus?

**Solution:**

6 abacuses * € 52,- = € 312,-

Sold for € 372,- dus € 372,- – € 312,- = € 60,- Profit

€ 60,- / 6 abacuses -= € 10,- per abacus

**3.** Leonardo wants to paint a wall that measures 5 by 3 m. A 2 litre can of white paint says that you need 1 litre of paint to paint 3 m2. How many cans must Leonardo at least buy to be able to paint the whole of the wall?

**Solution:**

Leonardo wants to paint 3 m * 5 m = 15 m^{2} white.

Usage 1 L paint per 3 m^{2}

1 can consists of 2 L paint

So with 1 can of paint he can paint 2 * 3 m^{2} = 6 m^{2} schilderen

3 potten = 18 m^{2} This is a little bit too much however half cans of paint are not sold so 3 is the correct amount.

**4**. A globe has been designed based on a scale of 1:15,000,000. The distance between two places is 9 cm on this globe. How large is the actual distance between these places?

**Solution:**

9 cm * 15.000.000 = 135.000.000 cm

135.000.000 cm = 1.350.000 m = 1.350 km

**5.** Leonardo earns € 15 an hour. He works 8 hours a day from Monday to Friday. He works 4 hours on Saturdays. How much does he earn a week?

**Solution:**

Leonardo works (5 * 8 hours) + 4 hours = 40 + 4 hours= 44 hours per week

44 * € 15,- = € 660,-

**6.** Leonardo wants to visit his parents who live on the other side of the forest. He can use a car and go by road or use a bicycle to cycle through the forest. The path through the forest up to his parents’ house is 5 km. The road around the forest to his parents’ house is 12 km long. Leonardo cycles at an average speed of 20 km/h. He drives at an average speed of 50 km/h when he uses a car. Which method will get him the fastest to his parent’s house? If he uses a car or a bicycle?

**Solution**

Bike: 5 km / 20 km/h = 0,25 hours

Car: 12 km / 50 km/h = 0,24 hours

So the car is fastest.

**7.** A car costs € 12,000. The price of the car is reduced by 25% when accessories are at least bought for a value of € 2990 when buying the car. The discount only applies to the car and not the accessories. Leonardo buys this car and € 3000 worth of accessories. What does he need to pay?

**Solution:**

Leonardo gets a 25% discount due to the accessories.

For the car he pays 0,75 * € 12.000,- = € 9.000,-

For the accessories he pays € 3000,-

Total = € 9.000,- + € 3000,- = € 12.000,-

**8.** Aristotle solves an equation in 3 minutes and 42 seconds. Pythagoras does this twice as fast. Leonardo needs 42 seconds more than Pythagoras. How much time does Leonardo need?

**Solution:**

Time Pythagoras = 3 min 42 sec / 2 = 1 min 51 sec

Time Leonardo = 1 min 51 sec + 42 sec = 2 min 33 sec

## Explanations Word Problems Medium

1. Leonardo sells an invention for € 1650. The customer can pay in 3 instalments. The first time he pays 2/5 of the total price. The second time he pays half of the remaining total. How much will Leonardo receive when the last payment is made?

**Solution:**

Payment 1: 2/5 * € 1650,00 = € 660,00

Residual debt after payment 1: € 1650,00 – € 660,00 = € 990,00

Payment 2 = Payment 3

Payment 3 = 1/2 * € 990,00 = € 495,00

2. Leonardo will be publishing his first book in 17 days. Today is Tuesday the 23rd of February. On which day will Leonardo be publishing his book?

**Solution:**

Starting day is Tuesday

Tuesday + 1 week is Tuesday

17 days is 2 weeks (2×7 days) + 3 days

Tuesday + 3 days is Friday

So on Friday.

3. Leonardo is looking for a catering company that can provide the food for this 50th birthday. He is given the following price estimates:

Ulti-Eat € 800 call-out charges + € 10 pp;

Munchers € 20 pp;

Uberchef € 650 call-out charges + € 12 pp;

La Cuisine € 1000 call-out charges + € 8 pp.

He is expecting 90 guests at his birthday party. Which catering company is the cheapest?

**Solution:**

Ulti-Eat: € 800,- + (90 * € 10,-) = € 1700,-

Munchers: 90 * € 20,- = € 1800,-

Uberchef: € 650,- + (90 * € 12,-) = € 1730,-

La Cuisine: € 1000,- + (90 * € 8,-) = € 1720,-

So Ulti-Eat is cheapest

4. How much is 6 m – 1 mm – 9 dm – 2 cm?

**Solution:**

Rewriting to mm gives:

6000 mm – 1 mm – 900 mm – 20 mm = 5079 mm = 5,079 m

5. Leonardo sees 23 boats in Venice. 2 boats have 14 people on-board each. 1/3 of the remaining boats have 4 people on-board. 3 people are on-board on each of half of the boats that have not yet been specified. 2 people are on each boat of the remaining ones. How many people are there on all boats when taken together?

**Solution:**

Total 23 boats

2 boats * 14 pers. = 28 persons

23 – 2 = 21 boats left

1/3 * 21 boats = 7 boats

7 boats * 4 pers. = 28 persons

21 – 7 = 14 boats left

½ * 14 = 7 boats

7 boats * 3 pers. = 21 personen

14 -7 = 7 boats over

7 * 2 = 14 persons

Total = 28 + 28 + 21 + 14 = 91 persons

6. Leonardo drives from Pisa to Milan at an average speed of 380 km/hour in his new Bugatti Veyron. It takes him 45 minutes. When he comes back, there is a traffic queue. The traffic queue is 15 km long and it takes him 1.5 hours to get through it. Next, an 80 km road section follows with slow traffic that drives at 40 km/hour. Because it is so busy, the average speed that he can attain when driving through the remaining section is only 190 km/hour. How long does it take Leonardo to drive back?

**Solution:**

Outward journey:

380 km/hour * ¾ = 285 km in 45 minutes

Homeward journey:

15 km in 1,5 hours

80 km with 40 km/hour so 80 km / 40 km/hour = 2 hours

Over: 285 km – 80 km – 15 km = 190 km with 190 km/hour = 1 hour

1,5 hours + 2 hours + 1 hour = 4,5 hours

7. Leonardo earns € 18 an hour. He works 8 hours a day from Monday to Friday. He works 6 hours on Saturdays and gets paid 150% of his normal wage. Leonardo works 4 hours in overtime every fortnight. He is paid 200% of his normal wage for overtime. How much does Leonardo earn a week?

**Solution:**

Mo – Fr he earns: € 18,- * 8 hours * 5 days = € 720,-

Saturday he earns: € 18,- * (150%)/(100%) * 6 hours = € 162,-

Per 2 weeks he works 4 hours in overtime.

On average per week he works 4 hours / 2 weeks = 2 hours overtime.

Overtime yields him per week : € 18,- * (200%)/(100%) * 2 = € 72,-

Total = € 720,- + € 162,- + € 72,- = € 954,-

8. Leonardo buys 3 notebooks that each cost € 3.40. He also buys 6 pencils and 3 rubbers. The pencils cost € 0.60 each. The rubbers cost € 0.50 each. Currently, the shopkeeper has an offer for a pencil and rubber bought together for € 1. How much does he have to pay in total?

**Solution:**

Notebooks: 3 * € 3,40 = € 10,20

Pencils together with rubbers: 3 * € 1,00 = € 3,00

Pencils separately: 3 * € 0,60 = € 1,80

Total: € 10,20 + € 3,00 + € 1,80 = € 15,00

9. Leonardo has bought a flock of sheep. He owns 60 + ¼ of his total quantity of sheep. How many sheep does Leonardo own?

**Solution:**

60 + ¼ for total amount of sheep

So 60 sheep = ¾ of his total amount of sheep

60 sheep / 3 = 20 sheep = ¼

Total = 4 * 20 sheep = 80 sheep

10. Leonardo has invented a new bicycle. He decides to test it on a 30 km track. He starts at a speed of 30 km/hour. He keeps it up for half an hour and, next, his speed drops by 5 km/hour. Every 12 minutes his speed drops by another 5 km/hour. How long will it take him to cover the track?

**Solution:**

First half hours: ½ hour * 30 km/hour = 15 km

So still 30 km – 15 km = 15 km to go

12 min = 1/5 hours so

15 km – (1/5 hours * 25 km/hour) = 15 km – 5 km = 10 km

15 km – (1/5 hours * 20 km/hour) = 10 km – 4 km = 6 km

15 km – (1/5 hours * 15 km/hour) = 6 km – 3 km = 3 km

15 km – (1/5 hours * 10 km/hour) = 3 km – 2 km = 1 km

15 km – (1/5 hours * 5 km/hour) = 1 km – 1 km = 0 km

so ½ hours + (5 * 1/5 hour) = 1 hour 30 min

11. Leonardo has a swimming pool that is 2 m high, 4 m wide and 6 m long. How many litres of water does he need to fill the swimming pool completely?

**Solution:**

Pool volume = 2 m * 4 m * 6 m = 48 m^{3}

1 L = 1 dm^{3}

48 m^{3} = 48.000 dm^{3}

So 48.000 liters of water

12. A bag contains 26 red, 52 black and 39 yellow marbles. Leonardo takes out 1 marble. What is the probability that he will take out a yellow marble?

**Solution:**

Total amount of marbles = 26 + 52 + 39 = 117

of which 39 yellow so the chance is 39/117 = 1/3

13. Today it is Friday the 18th of June. It will be Leonardo’s weeding anniversary in 45 days. On which date did Leonardo get married?

**Solution:**

June has 30 days

July has 31 days

18 june + 12 days = 30 June

30 june + 31 days = 31 July

Days left = 45 days – 12 days – 31 days = 2 days

31 July + 2 days = 2 August

## Explanations Word Problems Hard

1. Leonardo is heading out again in his Bugatti Veyron. Once more he drives from Pisa to Milan, at an average 380 km/hour. It takes him 45 minutes. On the return journey he gets stuck in traffic. The traffic jam is 10 km long, and it takes him 1.5 hours to get through. After this he drives another 60km in slow moving traffic at 40 km/hour, after which Leonardo can resume his route as normal. In total his average speed on both trips measures 114 km/hour. What was his average speed on the return journey when he wasn’t stuck in a traffic jam or slow moving traffic?

**Solution:**

Outward journey: ¾ * 380 km/hours = 285 km in 45 min (0,75 hours)

Return journey: 10 km in 1,5 hours

60 km / 40 km/hours = 1,5 hours so 60 km in 1,5 hours

Last part is 285 km – 10 km – 60 km = 215 km in x hours

Total time outward and return = 0,75 hours + 1,5 hours + 1,5 hours + x hours = 3,75 + x hours

Total time on full distance =570 km / 114 km/hours = 5 hours

3,75 hours + x hours = 5 hours

x = 1,25 hours

dus 215 km in 1,25 hours = 172 km/hours

2. Leonardo has 3 taps in his bath tub. If he only fully opens tap 1 it takes 10 minutes to fill the bath. If he only fully opens tap 2 it takes 15 minutes to fill the bath, and if he only fully opens tap 3 it takes 30 minutes to fill the bath. If Leonardo were to fully open all 3 taps how long would it take to fill the bath?

**Solution:**

To calculate this we need to use an arbitrary volume of the bath tub. This value doesn’t need to be realistic so lets take 10 liters.

Now we can calculate the amount of water running from the taps:

Tap 1: 10 L/ 10 min = 60 L/ hour

Tap 2: 10 L/ 15 min = 40 L/ hour

Tap 3: 10 L/ 30 min = 20 L/ hour

If all taps are opened this would mean 60 L/ hour + 40 L/ hour + 20 L/ hour = 120 L/hour will flow into the bath.

The volume of the bath tub is 10 L so (10 L / 120 L/hour) * 60 min = 5 min

3. Leonardo wants to make Invar. Invar is an alloy with an extremely low expansion coefficient, comprising 36% nickel and 64% iron. He has 20 kg of alloy comprising 20% nickel and 80% iron. And he has 180 kg of alloy comprising 50% nickel and 50% iron. How many maximum kilos of Invar can he make?

**Solution:**

Below is the calcolation based on the amount of Nickel

0,2 * 20 kg + 0,5 * x kg = 0,36(x + 20 kg)

4 + 0,5x = 0,36x + 7,2

0,14x = 3,2

x ˜ 23 kg (this is what he needs to ad)

In total he will end uop with 23 kg + 20 kg = 43 kg

4. Leonardo keeps ostriches and sheep: in total 80 animals. Together all the animals have 220 legs. How many ostriches does he have?

**Solution:**

80 animals

220 legs

x sheep

80-x ostriches

So the amount of legs gives 2*(80-x) + 4x = 220

– 2x + 4x = 220 – 160

2x = 60

x = 30

So 30 sheep and 80 – 30 = 50 ostriches

5. An aeroplane flies 800 km with the wind behind it. With the same capacity and in the same time it would fly into the wind at 720 km. The wind speed is 30 km/hour. What is the speed of the aeroplane without wind?

Set the speed of the plane without wind at v km/hours.

With the wind behind the plane v + 30 km/hours for 800 km

With the wind against the plane v – 30 km/hours for 720 km

Equation will be:

(v+30 km/hours)/800 km = (v-30 km/hours)/720 km cross-multiplication gives:

720v + 21600 = 800v – 24000

45600 = 80v

v = 570 km/hours

6. Leonardo wants to construct an ostrich run with 100 m fencing. Using this fencing he wants to make an ostrich run with as big a surface area as possible. The type of fencing that he uses cannot bend; but he does have 4 right-angled (90°) corner pieces. What will the surface area of this ostrich run be?

**Solution:**

Since he has 4 corner pieces of 90° and he can’t bend the fence he has to go for either a square or rectangular shape. It is a known fact that the biggest possible surface area with a known circumference is a square so 100 m / 4 = 25 m dus l = b = 25 m

If you don’t know this you can also use the following approach:

Circumference = 2(l + w) = 100 m

l + w = 50

l = 50 – w

Surface = l * w = (50 – w) * w = – w^{2} + 50w

This is a quadratic equation, when you put this in a graph, you will find a parabola of which you can pinpoint the top. The value at the y-axis at the top is maximum surface area, the value of the x-axis is the width at which the maximum surface area is reached.

There is also a possibility to do this without drawing a graph. We use the following formula for this: x_{parabooltop}= (-b)/2a

a en b can be derived fromk the quadratic equation:

Every quadratic has the following form: y = ax2 + bx + c

In this case that gives: opp. = -w^{2} + 50w

So a = -1 and b = 50

so w_{parabolatop} = (-50)/(2•-1)= (-50)/(-2) = 25 m

l + w = 50 m

l + 25 m = 50 m

50 m – 25 m = l = 25 m

Opp.max = 25 m * 25 m = 625 m^{2}

7. Leonardo usually pays € 0.50 for 3 pencils. He sells a pair for € 0.50. In one day Leonardo makes € 2.50 profit. How many pencils did he sell that day?

**Solution:**

The smallest common divisor of 2 and 3 is 6 so the solution looks like this:

He pays € 0,50 for 3 pencils, for 6 pencils he pays 2 * € 0,50 = € 1,00 6 pencils he sells for (6 / 2) * € 0,50 = € 1,50

profit per 6 pencils is € 1,50 – € 1,00 = € 0,50

He has a profit of € 2,50 so € 2,50 / € 0,50 = 5

So he sold 5 * 6 = 30 pencils

8. Leonardo divides € 14,000 out over 2 bank accounts. Bank account A earns 5% interest; bank account B 7%. At the end of the year Leonardo receives € 798. How much has Leonardo paid into bank account B?

**Solution:**

bank account B = x

bank account A = 14.000 – x

0,07x + 0,05(14.000 – x) = 798

0,07x – 0,05x + 700 = 798

0,02x = 98

x = € 4.900,-

So Leonardo stalled € 4.900,- on bank account B

9. Leonardo takes part in a cycle race. He sets off at a good speed of 35 km/hour. Once Leonardo has ridden half the race (in distance) he realises he set off too quickly and reduces his speed to 25 km/hour. He finishes the race at this speed. In total it takes him 3 hours and 36 mins. What distance has Leonardo cycled?

**Solution:**

Set half of the total distance at x. This leads to the following set of equations:

3 hours en 36 min = 3,6 hours

(x/35) + (x/25) = 3,6 hours

25x/875 + 35x/875 =3,6

60x/875 = 3,6

60x = 3150

x = 52,5 km

x was only half the distance so the total distance is 2 * 52,5 km = 105 km

10. Every Monday Leonardo cleans his toilet and every fifth day he vacuums. If today he is cleaning his toilet as well as vacuuming how long will it be before he cleans his toilet and vacuums on the same day again?

**Solution:**

The lowest common divisor of 5 and 7 can be found by multiplying –> 5 * 7 = 35

So in 35 days

11. Leonardo has designed a solar panel. He uses some of the power he generates with this solar panel for personal use, and the rest he sells to his energy supplier. Within the space of a month Leonardo uses 390 m3 gas and sells 130 KWh electricity. His fuel bill for that month amounts to € 130. The following month he uses 360 m3 gas and sells 160 KWh electricity. His bill for that month amounts to € 110. How much does the energy supplier pay Leonardo for 1 KWh electricity?

**Solution:**

Price 1 m^{3} gas = x

Price 1 KWh electricity = y

2 equations:

130 = 390x + 130y

110 = 360x + 160y

Rewrite equation 1 to:

130 = 390x + 130y

1 = 3x + y

y = 1 – 3x

Rewrite equation 2 to:

110 = 360x + 160y

110 = 360x + 160(1-3x)

110 = 360x + 160 – 480x

-360x + 480x = 160 – 110

120x = 50

x = 5/12

y = 1 – 3x

y = 1 – 3(5/12)

y = 1 – 1,25

y = -0,25

So Leonardo earns € 0,25 per KWh electricity.

12. Plato is approximately just as many days old as Aristotle is weeks old and Plato is just as many months old as Leonardo is years old. Together Plato, Aristotle and Leonardo are 140 years old. How old is Leonardo? (assume 1 year = 12 months = 52 weeks = 365 days)

**Solution:**

age Leonardo

age Aristoteles

age Plato

P + A + L = 140

A = 7P

L = 12P

completing this gives:

P + 7P + 12P =140

20P = 140

Plato = 7 years old

Filling in gives:

P + 7P + L = 140

7 + 49 + L = 140

L = 84

Leonardo is 84 years old

13. For his mathematics exams Leonardo has to take 4 tests. He achieves a score of 9.4 and 9.8 in the first 2. What average score must he achieve for the next 2 is he wants to achieve an average score of 9.5? The 2 tests he still has to take together account for 25% of the final score.

**Solution:**

Average now:

(9,4 + 9,8) / 2 = 9,6

This one counts for 75% so:

0,75*9,6 + 0,25x= 9,5

7,2 + 0,25x =9,5

0,25x = 2,3

x = 9,2

14. Think of a 3 digit number that when you switch round the ten and hundred numbers the resulting number is 20% more than the original number.

**Solution:**

100a + 10b + c =1,2 * (10a + 100b + c)

100a + 10b + c = 12a + 120b +1,2c

set c at 0

100a + 10b = 12a +120b

88a = 110b

Also known:

b < a, a < 10, b < 10

So simply try some and you’ll find:

a = 5 b = 4

88(5) = 110(4)

440 = 440

So 540 is the resulting number. We are looking for the original number so the answer is 450.

15. Leonardo wants to place bird cages in his garden. For this he has set aside 52 m^{2} of garden and € 825. Bird cage A has a capacity of 14 m3 and a ground area of 4 m^{2} and costs € 50. Bird cage B has a ground area of 8 m^{2}, a capacity of 30 m^{3} and costs € 125. How many of each bird cage does he need to place in order to attain as large a bird cage content as possible; within his budget and allocated space? The bird cages cannot be stacked due to their fragile construction.

First calculate the specific surface area(surface/volume) for both bird cages.

bird cage A: 4 m^{2} / 14 m^{3} = 2/7 = 30/105 m^{-1}

bird cage B: 8 m^{2} / 30 m^{3} = 4/15 = 28/105 m^{-1}

This shows that Leonardo wants to use as much bird cages of type B as possible since that one has the lowest specific surface area.

He has € 825 to spend:

€ 825 / € 125 = 6,6 so 6 bird cages of type B

He is left with € 825 – (6* € 125) = € 825 – € 750 = € 75 euro

Surface:

6 bird cages B * 8 m^{2} = 48 m^{2}

52 m^{2} – 48 m^{2} = 4 m^{2}

So he can also place 1 extra bird cage A of € 50 after which he ends up with € 50.

So:

bird cage A: 1 piece

bird cage B: 6 pieces

16. Leonardo is 9 years younger than 3x as old Archimedes. 4 years ago he was 10 years older than 2x as old Archimedes. How old is Leonardo?

**Solution:**

2 equations

L = 3A – 9

and

L-4 = 2(A – 4) + 10

This can be rewritten to:

3A – 9 – 4 = 2(A – 4) + 10

3A – 13 = 2A – 8 + 10

3A – 2A = -8 + 10 + 13

A = 15

Completing this gives:

L = 3 *15 – 9

L = 36 years