Math Problem : DD.32 Moving by running and a car

Hanako and Taro go to B point 12 km away from A point.
Hanako rides a car from A point first and getting down on the way, runs the remaining distance.
Taro runs from A point and rides the car on the way which took down Hanako and returned immediately on the way and goes to B point.
They leave at the same time and the speed of the car is 48 km/h.

(1) Since both of them ran with the same speed, they arrived at B point 30 minutes after leaving.
Find the distance from A point of the place where Hanako got off the car.
Moreover, find the running speed per hour of Hanako.

(2) If Taro runs with the speed of 12 km/h, how many minutes and how many seconds does he arrive at B point earlier than Hanako?

Math Problem : DD.31 Three persons from home to a park

Taro left home at 9:00 and ran to the park where is 14 km away from home at the speed of 200 m/m.
Whenever he ran for 10 minutes, he took a rest for 2 minutes.

(1) Find the time when Taro arrived at the park.

(2) The elder sister of Taro left home at 9:50 and she went to the park by bicycle without taking a rest on the way.
The elder sister arrived at the park 7 minute earlier than Taro.
Find the speed per minute of the elder sister.

(3) Taro’s father went back and forth between the house and the park by car at 60 km/h.
He took a rest three times on the way.
He took the 1st rest for 2 minutes together with the 2nd rest of Taro. He took the 2nd rest in the park.
He took the 3rd rest for 2 minutes together with the 5th break of Taro.
Find the time of the 2nd rest taken.

Math Problem : DD.30 Town on the top of mountain

There is A town on the top of the mountain and B town at the foot of the mountain. 

Hanako left A town and Taro left B town at the same time and they went back and forth one time without taking rest between A and B. 

They met first at intermediate C point. 

And they met again at D point where is 560 m from C point 74 minutes afterward. 

Hanako and Taro go up and down the mountain with the same speed. 

The ratio of the speed of going up and down is 5 : 7.

(1) Find the distance of A town and B town.

(2) Find the speed per minute of going up the mountain. 

Math problem : CC.19 Test of three problems

In a certain grade, there was a test of three problems A, B and C.
The percentage of the students who gave the right answer of A was 60%, who gave the right answer of B was 70% and who gave the right answer of C only was 15%.
Moreover, those who gave the right answer of A and did not give right answer of B were 18 persons.
Those who did not give right answers of all three problems were 5%.
Find the number of students of this grade.

Math Problem : C.32 Weight of log

The 5 m long log whose cut surface is the circle with radius 20 cm weighed 376.8 kg. 

Find the weight of 1 cm³ of this log. 

Math Problem : C.29 Sales number in three months

The sales number of a certain goods increased by 5% of the sales number of the last month in two consecutive months of June and July. 

The sales number in June was 420 pieces. 

In this case, how many sales number in July increase more than the sales number in May? 

Math Problem : JJ.57 Cut squares out of rectangular paper

I take squares out of the rectangular paper of which width is longer than length.

I take squares as many as possible whose one side is the shorter one of the rectangular paper first(Fig. 1).

If paper remains, I take squares as many as possible whose one side is the shorter one of the rectangular paper remained(Fig. 2).

This work is repeated and I finish the work when there is no remainder of the paper(Fig. 3).

(1) In the example when the length of the first rectangle is 16 cm in length, find the horizontal length.

(2) How many squares can be taken from the rectangle with 5 cm in length and 18.6 cm in width?

(3) When the length of the first rectangle shall be 10 cm in length, how many kinds of length considered as horizontal length is there in order to take five squares exactly?

Math Problem : C.25 Price of pork at butcher

At a certain butcher, the pork of which sales price is 1500 yen of 600 g was sold at a 15% discount. 

There are 25% of profits of the cost price when it is sold at this discounted price. 

Find the cost price per 100 g of this pork. 

A.3 Number of bottles filled with water

2 m³ of water is put into the plastic bottles with 1.8L capacity per bottle. 

In this case, find the number of the plastic bottles filled with water and the volume (mL) of the remaining water.

C.23 Sales change in three months

The sales in June of a certain store decreased by 15% compared with May. 

The sales in July of the store increased 12% compared with June. 

What percentage of reduction do the sales in July become compared with May? 

J.28 Paper tape around the equator of the earth

Find the area (km²) of the paper tape when an 8 cm wide paper tape is wound around the 
equator of the earth one round.
Noted that the earth is to be a sphere radius 6400 km.
The answer should be rounded off the 2nd decimal place.

J.21 Overlapped triangle and pentagon

The equilateral triangle and the equilateral pentagon have overlapped, as shown in the figure. 

Find the angle of the angle X.

EE.1 Two points moving on a diameter

The figure is a circle centering on the point O with radius of 6 cm.
The diameters AB and CD cross right-angled.
The point P leaves A and goes back and forth between A and O at 3 cm/s without stopping.
The point Q leaves B and goes back and forth between B and O at 2 cm/s without stopping.
P and Q leave at the same time.

(1) Find the area of the triangle CPQ 5 seconds after P and Q leave.
Moreover, find the numbers of seconds after leaving when it becomes the same area at next time.

(2) Find the area of the triangle CPQ 116 seconds after P and Q leave.

(3) There is a time when the area of the portion excluding the triangle CPQ from the circle is to be 107.04 cm2.
Find the number of seconds after P and Q leave when it becomes this area at 6th times.
Pi is assumed to be 3.14.

II.9 Connect red tapes and white tapes

There are a red paper tape and a white paper tape.

Red tape is cut to every 8 cm and white tape is cut to every 9 cm.

When I connected cut tapes by piling every 1 cm of the end of the cut tape with paste by turns so as to be different colors, it was 159 cm long.

Find the number of used sheets of red and white and the number of joint, respectively.

LL.22 Play at dice using figure

Play at dice using the figure.
When rolling dice and the spot of 1 or 2 comes out, move to the right by one.
When rolling dice and the spot of 3 or 4 comes out, move to the diagonal right by one.
When rolling dice and the spot of 5 or 6 comes out, move upwards by one.
When there is no way to move at rolling dice, remain at the place once.
This play sets out to the “ Goal “ from “ Start “.

(1) Taro rolled dice 5 times and made a goal without remaining from “ Start “.
In this case, find the most number considered in the sum total of the spot of a die and the least number, respectively.

(2) Next, two persons, Jiro and Hanako played. Hanako rolled dices 6 times to reach the place where Jiro reached rolling dice 3 times.
Choose the place from A to W.
What times is the least number of the sum total of the spot of the dice of Hanako is the least number of the sum total of the spot of the dice of Jiro considered at this time?

LL.21 Three persons take out red and white ball

There are ten red balls and ten white balls in the bag.

Three of Taro, Jiro and Hanako take out one ball at a time from this bag.

The taken-out ball is not returned to the bag.

When taken out at 1st time, the colors of the balls of Taro and Jiro are different and the colors of Jiro and Hanako are also different.

When taken out at 2nd time, the colors of Taro and Hanako are different and the colors of Jiro and Hanako are also different.

At this time, the color of Hanako is white and the color of Jiro is different from the color in the 1st time.

(1) When taken out at 2nd time, find the number of red balls and white balls in the bag respectively.

(2) As for at 3rd time, the person whose ball taken out at 3rd time was a red ball continues to take out a ball again.

When the 3rd time finished, one red ball remained in the bag.

Find the number of the remaining white ball.

KK.15 Accumulated cubes and paint

I accumulated cubes whose one side is 2 cm as shown in the figure and I painted the surface of this solid including the bottom.

And after paint got dry, I dissect it to pieces one by one.

(1) Find the number of the cube of which three faces are painted.

(2) Find the sum total of the area of the face which is not painted.

KK.11 Development view of rectangular prism

The character P is written to one face of a certain rectangular prism. 

Moreover, one diagonal line is drawn on each face of three faces. 

Fig. 1 and Fig. 2 shows two types of the development view of this rectangular prism. 

The length of the side AB is 14 cm.

(1) Draw three diagonal lines on Fig. 2 in handwriting.

(2) It is 24 cm when the length of the circumference of rectangle ABCD is subtracted from the length of the circumference (bold line) of the developed view of Fig. 1. 

Moreover, it is 10 cm when the length of the circumference of the developed view of Fig. 1 is subtracted from the length of the circumference (bold line) of the development view of Fig. 2. 

Find the volume of this rectangular prism. 

K.7 Assemble development view

The solid represented by the development view of the figure was assembled. 

When the developed view of A~D is assembled, which is that becomes the same pattern as this solid to be arranged? 

J.16 Partition and area of trapezoid

A figure is trapezoid ABCD whose side AD is parallel to the side BC. BE and AC are vertical. 

Find the area of this trapezoid.