Math Problem : DD.34 Speed ratio is 7 : 5

Taro and Jiro leave A point at the same time walk to B point.
The speed along which Taro and Jiro walk is constant respectively and the ratio of speed is Taro : Jiro = 7 : 5.
18 minutes after Taro arrived at B point, Jiro arrived at B point.

(1) After Taro arrived at B point, he rested for 10 minutes.
Then he went to A point with the same speed before resting.
How many minutes did it take after leaving B point for Taro to meet Jiro?

(2) Find the time which it takes when Jiro goes back and forth between AB points by 3 times faster than the speed Jiro walked.

Math Problem : HH.15 Number of ways from P to A

A figure shows the equilateral triangles whose one side is 1 cm are arranged.
The point leaves the point P and moves with the speed of 1 cm/s on the side.

(1) How many ways of going are there for the point to go to point A from point P in 3 seconds exactly without passing along the same point twice?

(2) How many ways of going are there for the point to go to point A from point P in 5 seconds exactly without passing along the same point twice?

Math Problem : DD.29 Notice an article left behind

There are 2400 m between A point and B point.
Taro left at 60 m/m and Jiro left at 50 m/m A point at the same time and they went to B point.
When Taro arrived at intermediate C point, it was 320 m between Taro and Jiro.

(1) Taro went to B point as he is.
When Jiro arrives at C point, find the distance with Taro.

(2) Taro noticed the thing left behind at C point and he returned to A point with the same speed.
Without resting he left A point at 270 m/m by bicycle and went to B point.
Jiro returned from B point without resting and went to A point with the same speed.

Find the distance from A point of the position where Taro and Jiro meet. 

Math Problem : KKK.14 Shadow of box by stand light

As shown in Fig. 1, the box of a rectangular prism is placed on a flat desk.
A stand 8cm in height is stood at right angle to the place 4 cm from the middle point of the side AB and the miniature bulb on the top of the stand is turned on.

Then, there is a shadow of the box as shown in Fig. 2.

Find the volume of the box.

Math Problem : HHH.7 Put 1~8 in a frame

Put one number from 1 to 8 in each frame.

Every number is put in so that it should be smaller than the number of right under and smaller than the number of right-hand side.

How many ways are there to put in numbers?

Math Problem : C.28 Janken and take marbles

Three persons, A, B, and C played the following game.

When they play a Janken (scissors-paper-rock) and only one person wins, the winner can get same number of marbles as marbles which the winner has from losers, respectively.

The game is over at the time someone's marbles are lost.

At first game A won and at following games, B and C in order won one time respectively. 

As a result, all marbles of A and B were lost. 

The number of the marbles which C finally had was 54. 

Find the number of marbles which A had first. 

Math Problem : JJ.56 Divide rectangle into three figures

Answer each of following questions.

(1) As shown in Fig. 1, when rectangle ABCD was divided into three figures, the area of three figures became the same. 

Find the length of EF at this time. 

(2) As shown in Fig. 2, when rectangle ABCD was divided into three figures, the area of three figures and the length of the circumference of three figures became the same, respectively.

Find the length of EG at this time.

Math Problem : LL.17 Seats of six persons

Six persons, A, B, C, D, E, and F, sat down groups of three face to face as shown in the figure.
These six persons explained how to sit down as follows.

A : I sat on the edge.
B : A was on the right hand side of the front looked at from me.
C : I sat on the front of E.
D : F was sitting on the left edge of the opposite side looked at from me.
E : C was sitting on the same side as F.
F : I and B were not facing each other.

Write the seat of A to F in a figure.

Math Problem : FF.7 Pass money among three persons

Three persons, Taro, Jiro, and Hanako, had some money, respectively.

Taro passed the half of money in hand to Jiro.

Next, Jiro passed 1/3 of the money in hand to Hanako.

Then, Hanako’s money in hand became the same as the money in hand which Taro had first.

Moreover, the ratio of the money in hand of Taro and Jiro became to 9 : 11.

When the total amount of money of three persons is 7600 yen, find the money in hand which Jiro had first.

GG.33 Price of consumption tax included

When we buy an article, consumption tax of 5% (less than 1 yen is omitted) of the price of the article is levied.

For example, since when you buy a 1250 yen article, a 62 yen consumption tax is levied, you should pay 1312 yen including a consumption tax.

Here, 1312 yen are defined as “the price of consumption tax included” of a 1250 yen article.


(1) Find the amount of the consumption tax of the article whose “the price of consumption tax included” is 5000 yen.

(2) There are improbable prices as “the price of consumption tax included”.
What is the least price among them?

Noted that “the price of consumption tax included” is 1 yen when the price of the article is 1 yen.


(3) Find the number of improbable prices as “the price of consumption tax included” among 5000 yen or less prices.

JJ.43 Four points on the side of rectangle

As shown in a figure, there are four points E, F, G and H on the side of rectangle ABCD.

(1) Find the area of the quadrangle EFGH.

(2) When point H is moved on the side CD in the direction to D, the area of the quadrangle EFGH became 30 cm². 

Find the length to which the point H was moved from the position of the figure.

JJ.42 Movement of rectangle overlapping triangle

As shown in a figure, there are rectangle ABCD and right triangle PQR. Rectangle ABCD moves at the speed of 1 cm/s to the right along the straight line XY from the position of a figure.

(1) Find the area of the portion with which the rectangle and the triangle have overlapped in 2 seconds.

(2) When it is in 3 seconds or more, the area of the portion with which the rectangle and the triangle have overlapped may become 2 cm². 

Find the time from the beginning of movement.

CC.15 Take out three kind of ball

There are three kinds of ball, A, B, and C and one piece of A, two pieces of B and three pieces of C are the same weight.
Moreover, when A, B, and C were mixed and 30 pieces were taken out, sum total weight was 232 g.
When they were divided for every kind of A, B, and C, the weight ratio of each group was 15 : 12 : 2.
Answer the following questions.

(1) Find the number of A, B, and C which were taken out, respectively.

(2) Find the weight per piece of A, B, and C, respectively.

CC.14 Exchange salt solution between two

There are two salt solution with different salt concentration in the two beakers A and B, respectively.
The ratio of the weight of the salt solution of A and B is 2 : 3.
The following two operation are done 1 time respectively in order.

- Move the half of the salt solution of A to B, and stir it well.
- Move the half of the salt solution of B to A, and stir it well.

The ratio of the salt concentration of the salt solution of A and B after two operations is to be 4 : 5.
Salt concentration of salt solution is the rate of the weight of the salt based on the weight of a salt solution.

(1) Find the ratio of the weight of the salt solution of A and B after two operations.

(2) When the salt concentration of the salt solution of A at first is 4%, find the salt concentration of the salt solution of B at first.

JJ.37 Fold and cut a rectangular paper

There is a paper of rectangle as shown in Fig. 1.

As shown in Fig. 2, this paper was folded so that the point B might overlap with the point D.

(1) What is the figure which will be made when ED and DF of Fig. 2 are cut with scissors and the remaining paper is opened?

(2) Find the area of the figure of (1).

JJ.34 12 points on a circumference

There is a circle with radius 6 cm as shown in the figure and there are 12 points are on the circumference arranged with the same interval.

The height of the equilateral triangle whose length of one side is 6 cm is assumed to be 5.2 cm and Pi is assumed to be 3.14.

(1) Find the length of AB.

(2) Find the area of the shadow portion.

F.6 Test scores of five persons

The test score of five persons A, B, C, D, and E, was as follows. 

- The average score of A, B, and C is 86 points. 

- The average score of C, D, and E is 77 points. 

- E’s score is 25 points lower than A’s. 

- The sum total of B’s and D’s score is 158 points. 

- A’s score is 10 points higher than C’s. 

(1) Find the score difference of B’s and D’s. 

(2) Find the C’s score.

F.5 55 pieces of one and five yen coins

There were one yen and five yen coins in total 55 pieces.

They were exchanged into ten yen coins so that the number of coins might decrease as many as possible.

The number of coins became 15 pieces and the number of one yen coins was 2.

The one yen coin which suited first should ask for number of sheets.

Find the number of pieces of one yen coin at first.

II.16 Four empty bottles for one new bottle

Juice is sold at 100 yen per bottle at a certain store and if you bring four empty bottles to the store, you can get one new bottle.

For example, you can drink nine bottles of juice if you have 700 yen.

(1) Find the number of bottle of juice which you can drink with 1900 yen.

(2) Find the fewest amount of money that you can drink 90 bottles of juice.

II.13 Sequences on a certain rule

As shown in a figure, the number is arranged under a certain rule.

Answer the following questions.

(1) Find the 10th number from the left in the 1st level.

(2) As for 32, find the number of level and the number counted from the left.