C.24 Discounted price with no loss

Taro bought in 150 goods whose one cost price is 3000 yen.

He started to sell goods at the fixed price with the expected profit of 60 percent of the cost price, but only 50 pieces were sold.

Then, he decided to make the price of remaining goods discounted by 20 percent of the fixed price and sell them.

Find the number of the goods at least which must be sold at the discounted price in order that Taro may not lose.

II.18 Number of teams are multiples of two in tournament

The tournament game in which number of teams participated is multiplied some 2 like 2, 4, 8, 16, -----, is considered.

It is expressed as N which is the number added all of several numbers that shows which round game is each game held on one tournament.

For example, when the number of teams is 8, the tournament becomes as shown in the figure.

As for the number of games, there are four games in the 1st round, two games in the 2nd round and one game in the 3rd round.

Therefore, when the number in ○ is added,

N = 1 + 1 + 1 + 1 + 2 + 2 + 3 = 11.

(1) Find N in the tournament whose number of teams is 16.

(2) Find N in the tournament in which the 6th round game is the final game.

(3) N was 4083 in the tournament of a certain number of teams.

In the tournament of twice as many number of teams as this, N is to be 8178.

Find the number of teams when N is 4083.

KK.21 Another cut line on the development view (KK21)

As shown in Fig. 1, there is cube ABCD-EFGH whose length of one side is 6 cm.

The points I and J are middle points of the sides AE and BF, respectively.

The points L and K are on the sides DH and CG, respectively and DL : LH = CK : KG.

This cube is cut by the plane which passes along four point I, J, K, and L, and it is divided into two solids.

Answer the following questions.

(1) When DL : LH = 1 : 2, find the volume of the solid including the vertex A.

(2) Draw the sides of the figure of the cut surface of (1) in the development view of Fig. 2.

(3) The ratio of the volume of a solid including the vertex A and a solid including the vertex E was 7 : 9.

Find the length of DL.

DD.42 Change direction at every meeting

Taro and Jiro leave P point at the same time and walk between P and Q point 90 m away from P.
Taro goes back and forth between PQ at 120 m/m.
Jiro walks at 15 m/m and every time after he meets Taro, he changes direction and walks.

(1) Find the distance from P point where Taro and Jiro met the 2nd time.

(2) Find the number of times that two persons meet in 12 minutes after leaving.

DD.41 Walk along riverside and ship in the river

There are P point where is upstream of the river and Q point where is downstream.
There is a ship which repeats going back and forth between PQ and this ship stops at P and Q for 18 minutes, respectively.
Taro walked along the riverside way from Q to P.
When Taro left Q, the ship left P at the same time.
Taro passed by the ship which goes to Q from P on the way twice.
Passing by the 2nd time was 1 hour 48 minutes after passing by the 1st time.
3 hours after leaving, Taro arrived at P and the ship arrived at Q at the same.
Answer the following questions.

(1) How many minutes did it take for Taro to pass by the ship which goes to Q from P at the 1st time after Taro leaves Q?

(2) Find the ratio of speed when the ship goes up the river and goes down the river.

(3) The speed of the ship without a flow is 132 m/m.
Find the speed of the flow of the river per minute.
Moreover, find the distance between PQ.

DD.40 Load dropped to a river from ship

There is a ship which moves at 45 m/m with a load and moves at 60 m/m without a load under no flow of the water.
This ship goes back and forth the river whose speed of a flow is 15 m/m.

(1) One day this ship carried the load from A point to downstream B point.
When the load was taken down at B point and it moved to downstream C point further, it took a total of 2 hours. It is 7.8 km from A point to C point.
Find the distance from A point to B point.
The time concerning taking down a load is not considered.

(2) On the other day, although the load has been dropped into the river as this ship is carrying the load toward downstream E point from D point, it moved as it is without noticing.
However, it noticed that it dropped before arriving at E point, it changed the direction and it went to the upper stream.
The load which has flowed from the dropped point were picked up, the ship changed the direction again and went to E point.
As a result, it took time 12 minutes more to carry the load than the schedule.
Find the time until the ship noticed dropping after dropping the load.
Time to change direction of the ship and time to pick up loads are not considered.