Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
(1) 23/7 - ( 6/5 × 7/3 / 2.8 + 1.7) =
(2) Find such a fraction when 5 is subtracted from a denominator, it will be 1/9 and when 5 is added to a numerator, it will be 1/4.
(3) The sequence of characters of "rikkyoniiza" is repeatedly arranged as rikkyoniizarikkyonizari --------.
Answer the following questions.
(3)-1 Answer the 50th character.
(3)-2 When counting only i, which number from the 1st character is the 2008th i ?
(4) Taro gets the pocket money of the same amount of money every month and also has some savings.
If he spends 2000 yen every month, the balance of savings come to be zero in six months.
Moreover, if he uses 1600 yen every month, the balance of savings come to be zero in nine months.
Answer the following questions.
(4)-1 Find the amount of the pocket money every month.
(4)-2 Find the amount of savings at the beginning.
(4)-3 In order for Taro to have the amount of savings 12000 yen in one year, find the amount of the pocket money which can be spent every month.
(5) As shown in a lower figure, a right hexagon is in a circle 3 cm in radius.
The point A is the middle point of the side.
Find the area of a shadow area.
Pi is assumed to be 3.14.
Problem 2
There is a running course of 3.2 km one round.
Taro and Jiro run to the same direction three round from the same position.
Taro began to run at 200 m/m and Jiro began to run at 240 m/m 3 minutes afterward.
When Taro finished running one round, he stopped and drank water.
While he was drinking water, he was passed by Jiro.
Then, Taro began to run at 260 m/m, he caught up with Jiro 3 minutes afterward.
Then, although Taro ran with the same speed as Jiro, when they finished running two round, Taro walked at 80 m/m.
Furthermore, after that, Taro began to run at 200 m/m and finished running three round later than Jiro for 7 minutes and 10 seconds.
Answer the following questions.
(1) Find the time while Taro was drinking water.
(2) Find the distance where two persons ran with the same speed.
(3) Find the distance where Taro walked.
Problem 3
As shown in a lower figure, the right hexagon DEFGHI is in equilateral triangle ABC.
Three points D, F, and H are on the side of the equilateral triangle and AD : DB = BF : FC = CH : HA = 2 : 3.
Answer the following questions.
(1) Find the ratio of the area of equilateral triangle ABC to the area of the triangle ADH.
(2) Find the ratio of the area of equilateral triangle ABC to the area of the right hexagon DEFGHI.
Problem 4
Numbers are arranged in a line as follows.
1/1, 1/2, 2/2, 1/3, 2/3, 3/3, 1/4, 2/4, 3/4, 4/4, 1/5, 2/5, -----
Answer the following questions.
(1) Find the 26th number.
(2) Find the sum total of the numbers from the 37th to the 55th.
(3) The sum total was 76 when I added numbers sequentially from the beginning.
Which number did I add by ?
Problem 5
Answer the following questions.
(1) How many ways of route from A to B is there in lower Fig. 1?
Noted that there are right or down directions of moving.
(2) How many ways of route from A to B is there in lower Fig. 2?
Noted that there are right or down or lower right directions of moving.
(3) How many ways of route from A to B with passing along neither P nor Q is there in lower Fig. 3?
Noted that there are right or down or lower right directions of moving.
Problem 6
There are white and black cubes whose one side is 1 cm.
63 white cubes and 62 black cubes are arranged in by turns as shown in lower Fig. 1 and a large cube is made.
Furthermore, this large cube is cut so that it may pass along three point A, B, and C.
Answer the following questions.
Noted that the volume of a triangular pyramid is calculated by the formula of (the area of the bottom) × (height) / 3.
(1) Draw the situation of the cut face in the lower Fig. 2.
(2) Find the ratio of the volume of a solid including the point D to the volume of the original large cube.
(3) Find the number of white cube in a solid including the point D which is not cut.
Problem 1
Answer
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
(1) 23/7 - ( 6/5 × 7/3 / 2.8 + 1.7) =
Answer
41/70
(2) Find such a fraction when 5 is subtracted from a denominator, it will be 1/9 and when 5 is added to a numerator, it will be 1/4.
Answer
3/32
Solution
N : D - 5 = 1 : 9
D - 5 = N × 9, D = N × 9 + 5
N + 5 : D = 1 : 4
D = (N+5) × 4 = N × 4 + 20
Thus N × 9 + 5 = N × 4 + 20.
N × 5 = 15, N = 3
D = 3 × 9 + 5 = 32
(3) The sequence of characters of "rikkyoniiza" is repeatedly arranged as rikkyoniizarikkyonizari --------.
Answer the following questions.
(3)-1 Answer the 50th character.
(3)-2 When counting only i, which number from the 1st character is the 2008th i ?
Answer
(3)-1 o
(3)-2 7361th
Solution
(3)-1
50 / 11 = 4 remainder 6 → 6th character = o
(3)-2
i is 3 pieces in one cycle.
2008 / 3 = 669 cycle remainder 1 → first i in one cycle is 2nd in one cycle.
669 × 11 + 2 = 7361th
(4) Taro gets the pocket money of the same amount of money every month and also has some savings.
If he spends 2000 yen every month, the balance of savings come to be zero in six months.
Moreover, if he uses 1600 yen every month, the balance of savings come to be zero in nine months.
Answer the following questions.
(4)-1 Find the amount of the pocket money every month.
(4)-2 Find the amount of savings at the beginning.
(4)-3 In order for Taro to have the amount of savings 12000 yen in one year, find the amount of the pocket money which can be spent every month.
(4)-1 800 yen
(4)-2 7200 yen
(4)-3 400 yen
Solution
(1)
9 - 6 = 3 months pocket money
14400 - 12000 = 2400 yen
1 month pocket money = 2400 / 3 = 800 yen
(2)
12000 - 800 × 6 = 7200 yen
(3)
12000 - 7200 = 4800 yen
4800 / 12 = 400 yen
800 - 400 = 400 yen
(5) As shown in a lower figure, a right hexagon is in a circle 3 cm in radius.
The point A is the middle point of the side.
Find the area of a shadow area.
Pi is assumed to be 3.14.
Answer
9.42 cm2
Solution
Shadow area can be transformed with same area to Fig.2 below.
3 × 3 × 3.14 × 120/360 = 9.42 cm2.
Problem 2
There is a running course of 3.2 km one round.
Taro and Jiro run to the same direction three round from the same position.
Taro began to run at 200 m/m and Jiro began to run at 240 m/m 3 minutes afterward.
When Taro finished running one round, he stopped and drank water.
While he was drinking water, he was passed by Jiro.
Then, Taro began to run at 260 m/m, he caught up with Jiro 3 minutes afterward.
Then, although Taro ran with the same speed as Jiro, when they finished running two round, Taro walked at 80 m/m.
Furthermore, after that, Taro began to run at 200 m/m and finished running three round later than Jiro for 7 minutes and 10 seconds.
Answer the following questions.
(1) Find the time while Taro was drinking water.
(2) Find the distance where two persons ran with the same speed.
(3) Find the distance where Taro walked.
Answer
(1) 35 seconds
(2) 2420 m
(3) 600 m
Solution
(1)
3200 / 200 = 16 minutes = Time for Taro to run one round
3200 + 260 × 3 = 3980 m = Distance for Taro to catch up with Jiro
3980 / 240 = 199/12 minutes = Time for Jiro to run
199/12 + 3 = 235/12 minutes = Time for Taro to run from start point
235/12 - 16 - 3 = 7/12 minutes = 35 seconds
(2)
3200 × 2 - 3980 = 2420 m
(3)
3200 / 240 = 40/3 minutes = Time for Jiro to run one round
40/3 + 7 minutes and 10 seconds = 41/2 minutes = Time for Taro to run last one round
(200 × 41/2 - 3200) / (200 - 80) = 15/2 minutes = Time for Taro to walk
80 × 15/2 = 600 m
Problem 3
As shown in a lower figure, the right hexagon DEFGHI is in equilateral triangle ABC.
Three points D, F, and H are on the side of the equilateral triangle and AD : DB = BF : FC = CH : HA = 2 : 3.
Answer the following questions.
(1) Find the ratio of the area of equilateral triangle ABC to the area of the triangle ADH.
(2) Find the ratio of the area of equilateral triangle ABC to the area of the right hexagon DEFGHI.
Answer
(1) 25 : 6
(2) 25 : 14
Solution
(1)
As shown in Fig.1,
the area of △ABC is assumed to be 5 × 5 = 25,
and the area of △ADH is assumed to be 3 × 2 = 6.
(2)
△DFH = 25 - 6 × 3 = 7
Right hexagon DEFGHI = △DFH × 2 = 7 × 2 =14 as shown in Fig.2.
Problem 4
Numbers are arranged in a line as follows.
1/1, 1/2, 2/2, 1/3, 2/3, 3/3, 1/4, 2/4, 3/4, 4/4, 1/5, 2/5, -----
Answer the following questions.
(1) Find the 26th number.
(2) Find the sum total of the numbers from the 37th to the 55th.
(3) The sum total was 76 when I added numbers sequentially from the beginning.
Which number did I add by ?
Answer
(1) 5/7
(2) 10.5
(3) 136th
Solution
(1)
| Group | 1st | 2nd | 3rd | 4th | 5th |
| 1/1 | 1/2, 2/2 | 1/3, 2/3, 3/3 | 1/4, 2/4, 3/4, 4/4 | 1/5, 2/5, -- |
1 + 2 + 3 + 4 + 5 + 6 = 21
Since the 26th number is 5th number in seventh group starting from 1/7, it is 5/7.
(2)
1 + 2 + 3 + 4 + 5 + 6 +7 + 8 = 36
37th = 1/9
55th = 10/10
1/9 + 2/9 + ------+ 9/9 + 1/10 + 2/10 + ----- + 10/10
= 45/9 + 55/10 =10.5
(3)
Sum of 1st group = 1
2nd group = 1.5
3rd group = 2
4th group = 2.5
5th group = 3
According to 1 + 2 + 3 + 4 + 5 + 6 +7 + 8 = 36, 76 - 36 = 40.
Then it is estimated that 1.5 + 2.5 + 3.5 + 4.5 + 5.5 + 6.5 + 7.5 + 8.5 = 40 and it is 40.
8.5 is sum of 16th group.
(1+16) × 16 / 2 = 136th
Problem 5
Answer the following questions.
(1) How many ways of route from A to B is there in lower Fig. 1?
Noted that there are right or down directions of moving.
(2) How many ways of route from A to B is there in lower Fig. 2?
Noted that there are right or down or lower right directions of moving.
(3) How many ways of route from A to B with passing along neither P nor Q is there in lower Fig. 3?
Noted that there are right or down or lower right directions of moving.
Answer
(1) 15 ways
(2) 41 ways
(3) 14 ways
Solution
Problem 6
There are white and black cubes whose one side is 1 cm.
63 white cubes and 62 black cubes are arranged in by turns as shown in lower Fig. 1 and a large cube is made.
Furthermore, this large cube is cut so that it may pass along three point A, B, and C.
Answer the following questions.
Noted that the volume of a triangular pyramid is calculated by the formula of (the area of the bottom) × (height) / 3.
(1) Draw the situation of the cut face in the lower Fig. 2.
(2) Find the ratio of the volume of a solid including the point D to the volume of the original large cube.
(3) Find the number of white cube in a solid including the point D which is not cut.
Answer
(1)
(2) 1 : 6
(3) 7 pieces
Solution
(1)
According to the figures below, cut face is to be the picture of answer.
(2)
The volume of a solid including the point D = 5 × 5 / 2 × 5 /3 = 125/6 cm3.
The volume of the original large cube = 5 × 5 × 5 = 125 cm3.
125/6 : 125 = 1 : 6
(3)
Solid line shows the cut line of upper face and dotted line shows the cut line of lower face.
Square with circle mark shows the cube not cut.
