The Fourth Grade Russian Mathematics Program

The Russian fourth grade textbook Mathematics: A Textbook for Grade 4 edited by A.I.

Markushevich, Ninth Edition, Moscow, 1980 was translated by UCSMP, and the translation is in their

archives. In this note we sample some of the key sections of this book and a number of the almost 1500

problems, many multi-part, that it contains. Our objective is to give a feel for the level of rigour, precision

and care with which the material is presented as well as the remarkable level of abstract reasoning that is

expected of the students. In this way we hope to give an idea of the level of problems and mathematical

material that fourth or fifth grade students are capable of handling. Of particular interest may be the

final section on more advanced problems from which we include a 20 problem sample.

Here is the table of contents:

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Our discussion starts in Chapter 1, Section 9, Sets of Any Elements with sample problems.135. There are 3 roads from Akulovo to Rybnitsa and 4 roads from Rybnitsa to Kitovo

(fig.45). How many ways can one travel from Akulovo to Kitovo by way of Rybnitsa?

136. One leg of a triangle is 3 decimeters, 6 cm long and another is 2 times longer. Thethird leg is 4 decimeters, 2cm shorter than the sum of the other two sides. Find theperimeter of the triangle.

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137. A piece of a receipt from a store was accidentally torn off (fig.46.). Reconstruct thereceipt.

138. Solve the equations:1) 69k = 14076;2) b× 74 = 22348;3) (7001 + x)× 42 = 441000;4) (8001 + y)× 32 = 656000.

142. For the anniversary of a school’s Young Pioneer organization, the students in grades6 and 7 planted 441 trees along the road. The seventh-graders planted 28 trees a dayand the sixth-graders planted 21 trees a day. How many days did it take the studentsto plant the trees if they all worked together? How many days did it take the studentsto plant the trees if the seventh-graders worked after the sixth-graders but each groupplanted the same number of trees as before?

163. The aurochs is one of the largest bovine animals in the world. This animal hasbeen entered in the Red Book (a list of endangered species). In 1940 5 aurochs werebrought to the Caucasus nature preserve. By the end of 1944 the number of aurochshad increased by 6, and by the end of 1956 the aurochs herd numbered 106 head. Byhow many head did the aurochs herd increase in the period from 1944 through 1956?

164. Solve the problems:1) A motorcyclist and a bicyclist are heading towards one another. They are now

272km apart. How many hours will it take them to meet if the bicyclist’s speedis 12km/hr and the motorcyclist’s is 56km/hr?

2) Two motorcyclists are heading towards one another. One is traveling at a speedof 62km/hr and the other is traveling at a speed of 54km/hr. How many hourswill it take them to meet if they are now 348km apart?

171. A young worker finished his job in 8hrs. He turned out 18 parts an hour. How muchtime will it take his foeman to finish the same work if he turns out 24 parts an hour?

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178. Can we compare numbers in which certain digits have been erased and replaced byasterisks?a) 72 ∗ ∗∗ and 75 ∗ ∗∗b) ∗3 ∗ ∗∗ and ∗8∗

180. Find the three-digit number ending in 1 which is greater than 981.186. Several 75 kopeck boxes of checkers were bought for a total of 9 rubles, 75 kopecks

and several 95 kopeck boxes of chess pieces were bought for a total of 6 rubles, 65kopecks. How many more boxes of checkers than of chess pieces were bought?

194. A pair of shorts costs 1 ruble, 45 kopecks and a T-shirt costs 1 ruble, 25 kopecks.Which costs more, 6 pairs of shorts or 8 T-shirts? How much more?

208. Solve the problems:1) 24 tons of coal were delivered to a school. 3/4 of the coal was used over winter.

How much coal was left?2) A painter use 4/5 of the paint he bought. How much paint was left if he bought

100kg?222. Solve the problem: “A vehicle transports a 3 ton load in one trip. How many tons

will the vehicle be able to transport in x trips if x equals 10 , 5, 0?227. For which values of the variables are the following equations true:

a) x = 4941× 198;b) y = 37, 920/12.

240. Solve the problems:1) A passenger train has 12 cars with 58 seats to a car. How many empty seats are

left if 667 people board the train?2) An auditorium seats 360. How many empty seats are left after 8 classes with 41

children to a class take their seats in the auditorium?Lesson on Expressions:

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247. The distance between two towns is 512km. How fast must a train travel in order tocover this distance in a hours? Write an expression and perform the operation for thefollowing values of a: 16, 8, 4, 2.

248. Daytime lasts a hours. How long does the night last? Write an expression. Give ananswer if a = 8, 10, 12.

249. x liters of milk were bought at 30 kopecks a liter and 3 liters of sunflower oil werebought at y kopecks a liter. What do the following expressions refer to? a) 30x; b)3y; c) x− 3; d) y − 30; e) 3y + 30x; f) y − 30x.

250. Ivan Ivanovich left home on a fishing trip to Bear Lake. He traveled 3 hours by trainat a speed of 75km/hr. He then walked xkm. How far was the trip from home toBear Lake?

251. A man is x years old and his son is y years old. The man is 30 years older than hisson. Complete the following table:

x 32 40 45y 1 5 30

x/y

How many times older was the man than his son when the boy was 1 year old, 5 yearsold, 30 years old? How many times younger was the boy than his father when theman was 32 years old, 35 years old, 45 years old?

260. I am thinking of a number. If it is increased by 15 and the result is multiplied by 8,160 is obtained. What number am I thinking of?

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Lesson on Equations:

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We continue with the problems following the lesson on double inequalities:371. There were five branches. Some crows flew over and sat in the branches. All the

branches were occupied, and no more than two crows sat on a single branch. Howmany crows could there have been? Write the set of numbers satisfying the problem’sconditions.

391. The length of a running track is 400m. In 6 min. 40 sec. Andrei ran 4 laps and Nikolairan 5 laps. How many meters per second faster is Nikolai’s speed than Andrei’s?

398. It took a power shovel 4 hours of work time to excavate 555m3 of earth. How manycubic meters of earth will a second power shovel excavate in 4 hours if it removes15m3 less earth per hour than the first.

400. A pumpkin was placed on one pan of a scale and several 2kg weights were placed onthe other. When 3 weights were placed on the pan the pumpkin weighed more andwhen 4 weights were placed on the pan the weights weighed more. What is the lowerapproximation of the pumpkin’s weight and what is the higher approximation? Writeyour answer in the form of a double inequality.

411. The length of a road (fig. 122) is known to be 27km from A to B, 17km from B toC, and 19km from A to D. Find the length of the road:a) from A to C;b) from D to B;c) from C to D.

415. The Ob River is 3,650km long, the Don is 1,870km, the Volga is 4,530km and theYenisei is 3,487km. Find the next higher and next lower approximations of the lengthsof these rivers that end in two zeros. Which approximation is closer to the length ofeach river?

419. Some fishermen caught 240kg of fish. 5/24 of the fish were perch and 7/12 were pike.Which did the fishermen catch less of, perch or pike?

434. Without adding, tell which sum is larger:3,251 + 785 or 3.521 + 7857,619 + 5,820 or 7,861 + 5,937.

436. Let A be the solution set to the inequality y ≤ 14/17. Which of the fractions 10/17,14/17, and 16/17 belong to set A? Express in set notation.

444. Two trains set out at the same time from Moscow and Rostov-on-the-Don and headedtowards one another. The train from Moscow traveled at a speed of 65km/hr and the

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train from Rostov-on-the-Don traveled 7km/hr more slowly. How many hours willit take the trains to meet if the distance between Moscow and Rostov-on-the-Don is1,230km?

454. One can travel by bus from Lidovo to Kuzminki with a change in Maryino or Sergeyevka(fig. 131). Which route is better if we know that: the trip from Lidovo to Maryinotakes 1 hr. 48 min., from Lidovo to Sergeyevka 1 hr. 25min., and from Sergeyevkato Kuzminki 1 hr. 35 min.? In Maryino there is a 5 min. wait for the bus, and inSergeyevka there is a 12 min. wait.

455. Segments AP and BM are equal (fig. 132). Are segments AM and BP equal?

469. A passer-by runs after his hat, which the wind carries away at a speed of 4m/sec.How will the distance between the passer-by and his hat change if he runs at a speedof 5m/sec.? How many seconds will it take him to reach his hat if they are now 9mapart?

474. A tap which pours 30 liters of water a minute was turned on and in 5 minutes abathtub was filled. The tap was then turned off and the drain was opened. The tubdrained in 6 minutes. How many liters of water were drained per minute?

483. 3/17 of a vegetable garden is planted with cucumbers, 5/17 is planted with tomatoes,and the rest of the garden is planted with potatoes. Which crop occupies the largerarea, cucumbers or tomatoes? What part of the garden is planted with potatoes?

492. Draw the diagram (fig. 146) in your notebook. Place numbers in the circles and

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numbers with operation signs on the arrows.

506. A truck left Leningrad for Moscow at 9 o’clock in the morning at a speed of 48km/hr.At 10 o’clock a car left Moscow for Leningrad at a speed of 82km/hr. How far apartwere the car and the truck at 12 o’clock on the same day if the distance betweenMoscow and Leningrad is 650km?

518. Solve the problems:1) 11 rubles 88 kopecks worth of sketchbooks and colored pencils were bought for

a group of kindergarten children. A sketchbook costs 30 kopecks and a box ofpencils costs 24 kopecks. How many children were there in the group if eachreceived one sketchbook and one box of pencils?

2) A child’s suit costs 36 rubles and an adult’s suit costs 95 rubles. A store sold1,517 rubles worth of suits. The same number of children’s suits and adult’s suitewere sold. How many children’s suits were sold?

541. Tolya began to read a book when Seryozha had already read 24 pages of the samebook. Will Tolya catch up to Seryozha in 5 days if he reads 18 pages a day andSeryozha reads 12 pages a day?

563. Find the area of a square if the perimeter equals 72cm.564. Find the perimeter of a square if the area equals 64cm2.572. Find the area of a rectangle with sides 3 times longer than the sides of a second

rectangle 16cm long and 12cm wide.579. Write the following statements in the form of an equality and determine the values of

the variable for which the equality will be true:1) the sum of x and 408 is 501 more than 312;2) 700 is 398 more than the sum of 45 and y;3) the difference of 145 and x is 533 less than 650;4) 611 is 89 less than the difference of 800 and b.

588. The side of a square equals mcm. Write expressions for calculating the perimeter andarea of the square and find the values for m = 35, m = 105.

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594. A collective farm raises cows, pigs, and sheep. The farm has 787 cows, x pigs, and1,213 more sheep than pigs. How many cows and sheep together does the farmhave? Solve the problem. Find the value of the resulting expression for x = 1, 914,x = 2, 213.

602. Apply the distributive law of multiplication:

a) (68 + x) · 2 f) 7m− 7 · 20b) (m− 7) · 5 g) 6 · 12 + 6xc) 17 · (14− a) h) 12 · 5− 12xd) 13 · (2 + m) i) 18k − 9 · 3e) 4a + 4 · 13 j) 7p + 11p

603. For what values of the variable is the equality true:

a) 3 · (x + 5) = 3x + 15 d) (5− 3)x = 5x− 3xb) (3 + 5)x = 3x + 5x e) (5− 3)x− 5x− 3 · 2c) (7 + x) · 5 = 7 · 5 + 8 · 5 f) (x + 2) · 4 = 2 · 4 + 2 · 4?

612. A milk can contained 36 liters of milk. After 4 liters were poured into another milkcan, each can contained the same amount of milk. How much milk was there in theother can originally?

A sample lesson:

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644. Solve the following equations:

a) (x + 59)/42 = 86 d) 43k − 215 = 473b) 6528/(a− 39) = 64 e) 89x + 68 = 9057c) m/38− 76 = 38 f) 5905− 27p = 316

647. A table is 9 times more expensive than a chair. Together they cost 40 rubles. Howmuch does the chair cost? How much less does the chair cost than the table?

650. In preparing a compound for polishing copper-ware, 10 parts water, 5 parts liquidammonia, and 2 parts chalk (by weight) are used. How many grams of each substancemust be used to prepare 390g of the compound?

654. In making cherry preserves , 3 parts sugar is used to 2 parts cherries (by weight).What amount of sugar and cherries is used for preserves if 7kg 600 g more sugar isused than cherries?

673. 11 parts water, 5 parts liquid ammonia, and 4 parts casein (by weight) are used tomake casein glue. How much glue is obtained if 60g less liquid ammonia than wateris used?

675. A library needs to have 1,800 books rebound. One shop can do the job in 3 days andanother can do the job in 6 days. How many days will it take both shops to rebindall the books if the work simultaneously?

676. Replace the number 1,800 in problem 675 by 960. Solve the new problem. What canyou conclude?

685. Write an equation to solve the problems:1) Michael has twice as many nuts as Nikolai, and Peter has 3 times as many nuts

as Nikolai. How many nuts does each have if they have 72 nuts in all?

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2) Three girls collected 35 seashells on the beach. Galya found 4 times as manyshells as Masha, and Lena found twice as many shells as Masha. How manyshells did each girl find?

704. Perform the operations in the diagram (fig. 186) and explain the result.

713. Two teams of collective farm workers working together stored 1,200 t of silage. Eachday one team stored 40 t of silage and the other stored 35 t. How much silage dideach team store?

714. One worker worked 7 hours a day for 3 days and another worked 8 hours a day for 2days. Together they produced 481 items. How many items did each worker produceif both worked at the same rate of output?

716. Write an equation to solve the following problems:a) A customer spent 4/5 of his money on a suit. How much money did the customer

have originally if the suit cost 60 rubles?b) Before lunch 7/10 of the grain in a freight care was unloaded. How many tons of

grain were there originally in the freight car if 42 t were unloaded?724. 36 students went to see a movie. This was 6/7 of the class. How many students were

there in the class?725. The area of a rectangle is 616 cm2 and its length is 28 cm. Find the area of a square

with a perimeter equal to the perimeter of the rectangle.Lesson on division with remainder:

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The next lesson is on common divisors and multiples:750. A boy was given 90 kopecks and asked to spend it all o one kind of candy. A snack

bar was selling candy at 2 kopecks, 5 kopecks and 11 kopecks a piece. Which candiescould the boy buy?

751. An iron bar weights 16kg. How many bars will it take to cast 41 machine partsweighing 12 kg. each? (Remember that the bars must be entirely melted to cast themachine parts.)

753. The perimeter of a rectangle, the length of whose sides can be expressed in integers,is 12 cm. How many different rectangles are there with this perimeter? Which ofthese has the largest area?

760. Write the integers from 1 to 2o in increasing order and underline in read pencil everysecond number and in blue pencil every fifth number. Which numbers are underlinedin red and which are underlined in blue? Which numbers are underlined in bothcolors? Which numbers are not divisible by either 2 or 5?

764. Write all the three-digit numbers with digits consisting only of 0, 2, and 5, and whicha) are divisible by 2;b) are divisible by 5.

769. Solve the problems:1) A girl read 25 pages of a book and then another 10 pages of the same book. In

all she read 7/10 of the book. How many pages are there in the book?2) A driver used 55 liters of gasoline and then another 35 liters. In all he used 9/10

of the tank of gasoline. How many liters of gasoline were in the tank originally?

The lesson on division by 3 and by 9:

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The next lesson is on fractions:

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808. A log 4 23 m long was sawed into pieces 1

3 m long. How many pieces were obtained?812. Write an equation to solve the problem:

1) The perimeter of a rectangle is 42 cm. It is 6 times longer than it is wide. Findthe area of the rectangle.

2) A rectangle is 5 times longer than it is wide. Find the area of the rectangle if itsperimeter is 72 cm.

819. Some apples were delivered to a warehouse on two trucks. One truck carried 4 310 t

of apples and the other carried 1 110 t less. How many apples were delivered to the

warehouse?821. A forester traveled 3 km on foot and rode 4 hours on horseback. What was the horse’s

speed if the entire distance covered was 34 km?831. Compare:

a) 3/4 and 5/8 using parts of a circle;

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b) 2/5 and 3/10 on a ray.852. Two men set off on foot simultaneously from the same point in opposite directions.

In 3 hours they were 21 km apart. Find the speed of the second man if the speed ofthe first was 4 km/hr.

853. A bicyclist left town A for town B. Three hours after he left a motorcyclist left townB heading in the direction of the bicyclist at a speed of 42 km/hr. 2 hours later theymet. Find the bicyclist’s speed if town A and B are 144 km apart.

868. There are a number of passengers seated in a Volga automobile. The car by itselfweighs 1,125 kg more than its passengers. What is the weight of the car alone if itweighs 6 times more than the passengers?

875. Three zeros were added to the right of the number 608. By how many ones did thenumber increase? By what factor did the number increase?

883. The longest day in Moscow lasts 1,053 minutes. Express the duration of this day inhours. What is the duration of the shortest night?

913. Arrange in increasing order : 3.456; 3.465; 8.149; 8.079; 0.453. Arrange the samenumbers in decreasing order.

916. Between what two successive integers are the following fractions?

a) 2.7; b) 12.21; c) 3.343; d) 9.1111?

924. Compare:

a) 3.573 and 3.581; d) 6.504 and 6.505;b) 8.605 and 8.59; e) 3.29 and 3.3;c) 7.299 and 7.3; f) 4.85 and 0.1;

Expanded form for decimals:

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931. What numbers correspond to points A, B, C, D and K on the ray (fig. 213):

936. Petya had an equal number of beetles and spiders in his collection. The beetles andspiders had a total of 308 legs. How many spiders and how many beetles were therein Petya’s collection.1

940. How many degrees does an angle contain if it forms:a) 1/2 of a right angleb) 1/3 of a right anglec) 1/2 of a straight angled) 1/3 of a straight angle

947. Petya spent 1 hour 40 minutes doing his homework. It took him 1/5 of this timeto solve a problem and 1/4 of the remaining time to solve an equation. How manyminutes did it take Petya to solve the problem and the equation?

957. Solve the problems:1) A large pair of scissors costs twice as much as a small pair. A small pair costs 80

kopecks less than a large pair. How much does each pair of scissors cost?2) A rooster weighs 5 times less than a turkey, and the turkey weighs 8 kg more

than the rooster. How much does each bird weigh?962. Volodya, Petya, and Tolya collected a total of 210 kg of paper for recycling. How

much paper did each collect if Petya collected twice as much as Volodya, and Tolyacollected twice as much as Petya?

1 Hint for U.S. students: Spiders have eight legs and beetles have six legs.

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Adding Decimal Numbers:

968. A parachutist fell for 4 seconds before opening his parachute. He fell 4.9 m thefirst second, and each of the following seconds he fell 9.8 m further than he fell thepreceding second. How many meters did he fall in 4 seconds?

976. The first runner in a relay race ran 100 m in 10.2 seconds, the second ran 200 min 20.5 seconds, the third ran 400 m in 46.7 seconds and the fourth ran 800 m in 1minute, 46.7 seconds. How long did it take the runners to run the entire distance?What was the length of the race?

982. A bicyclist’s speed is 15 km/hr and a pedestrian’s speed is 9.7 km/hr less. At whatspeed would they approach each other from opposite directions?

984. A river boat’s actual speed (in calm water) is 21.6 km/hr. The speed of the currentis 4.9 km/hr. Find the boat’s speed upstream and downstream.

993. Solve the equations:

a) 8.6− (x + 2.75) = 1.85; b) 93− (71.8− y) = 22.48.

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994. Write four numbers of which the first is 4.612 and each succeeding number is 2.154greater than the preceding one.

1000. Without calculating, state which sum is greater:a) 3.72 + 0.89 + 5.21 or 3.84 + 0.98 + 5.64b) 21.4 + 8.3 + 6.1 or 20.7 + 8.1 + 5.73

’ Multiplying Decimal Numbers:

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1023 Using the variables x and y, write in general terms the commutative law of multipli-cation. Check it for:

a) x = 2.8, y = 1.05; b) x = 17, y = 0.17.

1024. Using the variables a, b, and c write in general terms the associative and distributivelaws. Check them for a = 8.5, b = 10, c = 0.2.

1026. Multiply:a) 6.25 · 4.3; h) 3.45 · 160;b) 85 · 8.8; i) 0.25 · 480;c) 9.4 · 38; j) 0.98 · 1.74;d) 12.6 · 7.5; k) 6.023 · 5.6;e) 0.8 · 9.75 l) 3.4 · 18.478;f) 250 · 2.31; m) 2.749 · 0.48.

1036. Use the distributive law to find the value of the expressions:a) 57.48 · 0.9093 + 42.52 · 0.9093;

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b) 6.395 · 835.67 + 6.395 · 164.33l;c) 104.76 · 378.91− 94.76 · 378.91;d) 0.78 · 496.6− 396.6 · 0.78.

1042. 1.5 kg of butter and 1.8 kg of sausage were purchased. How much was spent if thesausage cost 2.5 rubles per kg and the butter cost 1.1 rubles more than the sausage?

1052. Write the set of numbers representing the sums of all the possible pairs of 2.38, 3.52,4.8, 6.02. Label the resulting set M . Which of the numbers 7.32, 8.4, 10.1, 5.8 belongto set M and which do not?

1058. I am thinking of a number. If the same number plus 2.5 is added to this number theresult is 12.5. What is the number?

1064. A school corridor is 30.24 m long and 5.12 m wide. Find the area of the corridor.Round the answer off to the nearest hundredth.

1067. I traveled from Zarechnaya to Mukhino by foot in 0.8 hours at a speed of 5.5 km/hrand from Mukhino to Kamenka by bicycle in 1.4 hours at a speed of 12.5 km/hr. Howmuch farther is Mukhino from Kamenka than from Zarechnaya?

1070. Simplify the following expressions:

a) 8.3a + 1.7a; c) 2.5x + 1.2 + 3.6x + 5;b) 71.4y − 70.2y; d) 8.8− 9.7a− 2.5a− 3.7.

1072. Solve the equations:a) 45.7x + 0.3x− 2.4 = 89.6;b) 80.1m− 10.1m + 4.7 = 74.7.

1075. Find the value of the expression 61.3x if x = 8; 10; 35; 100; 1,000; 10,000.1084. Vanya spent four days collecting mushrooms. The first day he found 8 white mush-

rooms, and on each of the following days he found 1.5 times more mushrooms thanon the preceding day. How many white mushrooms did Vanya collect in 4 days?

1103. Find the weight of 23 identical machine parts if it is known that 12 such parts weigh43.2 kg.

1104. 3 kg of grapes, at 1.2 rubles a kilogram, and 7 kg of pears were bought. Twice asmuch was spent for the pears as for the grapes. How much does 1 kg of pears cost?

1110. Express in decimal form:

a) 34 ; b) 5

8 ; c) 74 ; d) 23

25 ; e) 5 12 ; f) 70 3

75 .

1129. Find the value of the expression 40.8a if a = 10; a = 100; a = 1, 000.

1133. A customer had 72 rubles. He bought a cap and a tie. He spent 0.1 of his money onthe cap and 0.01 of his money on the tie. How much money did he have left?

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Lesson on percents:A hundredth part of a ruble is called a kopeck, a hundredth part of a meter is called

a centimeter, and a hundredth part of a hectare is called an are.2 We often need to dealwith hundredth parts of various quantities: monetary sums, weight, volume, and so on.In many cases a single word is used for a hundredth part - percent. Thus 1 kopeck is onepercent of a ruble, 1 cm is one percent of a meter, 1 are is one percent of a hectare, 5percent of a ruble is 5 kopecks, 10 percent of a meter is 10 cm, 11 percent of a hectare is11 ares.

A hundredth part is called a percent.The word “percent” following a number expressed in digits is replaced by the symbol

% . For example the statement “1% of our school’s Young Pioneers were sent to a rally,”“2% of the area of the garden was sown with onion seed” are read: “One percent of ourschool’s Young Pioneers were sent to the rally”; “Two percent of the area of the gardenwas sown with onion seed.”

1139. 850 kg of cucumbers were delivered to a market stall. One customer bought 1% of thecucumbers, and a second customer bought 3% . How many kilograms of cucumbersdid each customer buy?

1142. What percent of a number is half that number? one fourth that number? Whatfraction of a number is 20% of that number?

1148. I was given 50 kopecks. I sent 76% of this money on food in the school snack bar,and later spent the rest on ice cream. Hom much did the ice cream cost?

1170. 220 horses and colts were driven to a watering hole. The colts made up on 15% ofthe herd. How many colts were there?

1174. When dried, mushrooms lose 79% of their weight. What amount of dried mushroomscan be obtained from 20 kg of fresh mushrooms?

2 A hectare is 10,000 square meters or the area of a 100 m × 100 m square, while an are is the area

of a 10 m × 10 m square.

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Decimal division:

1197. A man’s stride is 0.8 m. How many strides must he make in order to walk a distanceof 100 m?

1198. The unit used in weighing gems is the karat. One karat equals 0.2052 g. Find theweight in karats of a diamond weighing 0.7182 g.

1201. Find the root of the equation:a) 1000− 0.708p = 999.57166;b) (x + 26.1)2.3 = 70.84l;c) (x− 1.2)/0.604 = 21.14;d) (10.49− a)/4.02 = 0.805:e) 8.2x− 4.4x = 38.38;f) 9a = 8.67a = 0.6699.

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1205. How many kilograms less does 1 m3 of cork weigh than 1 m3 of water if 1 cm3 ofwater weighs 1 g and 1 cm3 of cork weighs 0.22 g?

1217. A motorboat travels the 210-km distance between two river ports in 5 hours headingdownstream and in 6 hours heading upstream maintaining the same actual speed atall times. Find the boat’s actual speed and the velocity of the current.

1227. A shipment of 24,000 bricks was delivered to a construction site. Breakage totaled2.5% of the shipment. How many unbroken bricks were delivered to the constructionsite?

1233. It is more than 10,000 km from Brest to Vladivostok. Will a map of the USSR, drawnto a scale of 1:10,000,000 fit on a page in your notebook?

1285. Ice cream contains 14% sugar. How much ice cream is produced if 35 kg of sugar isused?

1288. Three boys collected a sum of money and bought a soccer ball. One boy contributed25% of the sum, another contributed 40%, and the third contributed the rest. Thefirst contributed 1 ruble. How much did the third boy contribute? How much did theball cost?

1302. A 5% saline solution is to be prepared. How much water and how much salt will ittake to produce 8 kg of solution?

1337. 3.4 kg of cookies and 0.85 kg of candy were bought for 7.14 rubles. What is the priceof 1 kg of cookies and 1 kg of candy if the cookies cost 2.5 times more than the candy?

1338. There were 3.2 t of rice at a warehouse. 80% of the rice was shipped to a store and toa market. 1.28 t more rice was shipped to the store than to the market. How muchrice was shipped to the market?

1339. 1,030 kg of beef was delivered to a cafeteria. The beef contained 20.6% protein. Howmany kilograms of protein did the beef contain?

1357. A rectangular parallelepiped is a cm long, 2 cm narrower than it is long, and 5 cmhigh. Write an expression for the volume of the parallelepiped.

1360. When they were 600 m apart, Seryoxha set off on his bike to catch up with Natasha,who was walking. In 4 minutes Seryozha caught up with Natasha. How fast wasNatasha walking if her speed was 4 times less than Seryozha’s?

1368. A student spent 4 times longer preparing his lessons than watching cartoons on tele-vision. How long was the cartoon program if it was 30 minutes shorter than thehomework preparation?

1370. A book contains three stories. The first story is 4 times as long as the second, andthe third is twice as short as the first. How many pages long is the book if the secondstory is 3 pages shorter than the third?

1374. A boy is 2/11 as old as his father and is sister is 5/11 as old as her father. How oldis the father if the combined age of the boy and the girl is 28?

1382. A father is 40 years old and his son is 5 years old. How many times older than hisson will the father be in two years? How much older than his son is the father now,and ho much older than his son will he be in two years?

1388. If a certain number is decreased by 15% and then increased by 14% of the original

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number we obtain 999,900. What is the number?1397. I am thinking of a number. If it is decreased by a factor of 4 and the result is increased

by 23, we obtain 38. What number am I thinking of?Problem 1397 is the last problem in the main body of the book.

The final set of problems is entitled More Advanced Problems

1398. Write the largest ten-place number in which all the digits are different.1399. Write the smallest ten-place number in which all the digits are different.1405. How many ways can the number 50 be represented as the sum of two even numbers?

Notations which differ only in the order of addends should be considered identical.1409. What is the final digit of the product of all integers from 1 to 81?1410. How many zeros are there at the end of the product of all integers from 10 to 25?1411. A cashier has 5-kopeck pieces and 10-kopeck pieces. How many ways can he give 50

kopecks in change?1412. Seven players participated in a chess tournament. Each played one game with every

other player. How many games did they play in all?1413. Each pair of twenty cities is connected by a non-stop air route. How many routes are

there in all?1414. Find the sum of all three-digit numbers which can be written using 1, 2, and 3 so

that in each number all the digits are different.1415. A group of 100 tourists arrived in a city. Of these, ten knew neither French nor

German, 75 knew German, and 83 knew French. How many knew both French andGerman?

1418. One container holds 8 liters, a second holds 5 liters, and a third holds 3 liters. Thefirst container is filled with water and the others are empty. How can these containersbe used to measure out 1 liter of water? 4 liters?

1419. There are 7 full barrels, 7 half-empty barrels, and 7 empty barrels in a basement.How can these barrels be loaded on three trucks so that each truck carries 7 barrelsand all three trucks carry the same load?

1422. Two trains, each 80 cars long, meet on a single track with a small dead-end siding.How can these trains pass if the siding can hold a locomotive and forty cars? (Thetrains can move backwards.)

1425. there are five lengths of chain, and each has 3 links. How can these be joined into asingle chain if only 3 links are opened and riveted shut?

1426. There are 60 fourth-graders in a certain school. Show that at least two of the celebratetheir birthdays the same week.

1432. Three friends meet: the sculptor Blondin, the violinist Brown and the artist Redman.“It’s remarkable that one of us has blond hair, one brown hair, and the other redhair, but no one’s hair is the color suggested by his surname.” the dark-haired friendremarked. “You’re right,” said Blondin. What color hair does the artist have?

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1440. The sum of two numbers equals 462. One of these ends in a zero. If this zero isremoved, we have the second number. What are these numbers?

1444. If a student buys 11 notebooks he will have 5 kopecks left. If he tries to buy 15, hewill be 7 kopecks short. How much money does he have?

1449. Two students each decided to buy a copy of the same book. One of the students was1 kopeck short, and the other was 42 kopecks short. When they pooled their moneythey still didn’t have enough to buy even one copy of the book. How much did thebook cost?

1450. One casting yields 6 machine parts. The wast from 6 castings allows for one morecasting. How many machine parts can be produced from 36 castings?

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