Time : 50 minutes
Passing mark : 70 %
Answer : End of the problem
Problem 1
Problem 2
Problem 3
Problem 4
Problem 7
Problem 8
Problem 10
Calculation
126 / 72 + (16 - 3) / 4 =
Problem 2
Calculation
1 - (2/3 - 0.16) / 19/25 =
Problem 3
Find X.
1/6 + 7/3 / X × 5/7 = 1
Problem 4
The sales in June of a certain store decreased by 15% compared with May.
Problem 5
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?
Problem 5
The equilateral triangle and the equilateral pentagon have overlapped, as shown in the figure.
Problem 6
Find the angle of the angle X.
Problem 6
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.
Noted that the earth is to be a sphere radius 6400 km.
The answer should be rounded off the 2nd decimal place.
Problem 7
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.
Problem 8
If water is put into a certain tank only with A pipe, it will be filled to the brim with water in 15 minutes.
If water is put in only with B pipe, it will be filled to the brim with water in 10 minutes.
When water was put into this tank only with B pipe first and water was put in with both pipes after that, it was filled to the brim with water in 7 minutes.
Find the time (minute) when put in water only with B pipe.
Problem 9
Taro's father is two years older than his mother.
The age of Taro's elder brother is 1/3 of his father's age.
The age of Taro will become 1/3 of his mother's age in five years.
Find the age difference of Taro and his elder brother.
Problem 10
The character P is written to one face of a certain rectangular prism.
Problem 11
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.
Problem 11
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.
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.
Problem 12
The figure is a circle centering on the point O with radius of 6 cm.
Problem 13
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.
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.
Problem 13
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.
<Answer>
Problem 1
Problem 2
Problem 3
Problem 4
Problem 7
Problem 8
Problem 10
Calculation
126 / 72 + (16 - 3) / 4 =
Answer
5
Problem 2
Calculation
1 - (2/3 - 0.16) / 19/25 =
Answer
1/3
Problem 3
Find X.
1/6 + 7/3 / X × 5/7 = 1
Answer
2
Problem 4
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?
Answer
Problem 5
4.8％
Solution
The sales in May = 100
The sales in June = 100 × (1 - 0.15) = 85
The sales in June = 85 (1 + 0.12) = 95.2
100 - 95.2 = 4.8
Problem 5
The equilateral triangle and the equilateral pentagon have overlapped, as shown in the figure.
Find the angle of the angle X.
Answer
Problem 6
85 degrees
Solution
One angle of right hexagon = 108 degrees
108 + 60 + (108 - 83) = 265
360 - 265 = 95
180 - 95 = 85 degrees
Problem 6
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.
Noted that the earth is to be a sphere radius 6400 km.
The answer should be rounded off the 2nd decimal place.
Answer
3.2 km2
Solution
0.8 cm = (0.08 m / 1000) km
The length of the equator = 6400 × 2 × 3.14 = 12800 × 3.14 km
The area of the paper tape = 12800 × 3.14 × 0.08 / 1000
= 128 × 8 × 3.14 / 1000
= 3.21536 km2
Problem 7
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.
Find the area of this trapezoid.
Answer
462 cm2
Solution
The height of the trapezoid ABCD = 35 × 18 / 30 = 21 cm
(14 + 30) × 21 × 1/2 = 462 cm2
Problem 8
If water is put into a certain tank only with A pipe, it will be filled to the brim with water in 15 minutes.
If water is put in only with B pipe, it will be filled to the brim with water in 10 minutes.
When water was put into this tank only with B pipe first and water was put in with both pipes after that, it was filled to the brim with water in 7 minutes.
Find the time (minute) when put in water only with B pipe.
Answer
2.5 minutes
Solution
The volume of the tank is assumed to be 150.
The volume of water per minute from A pipe = 10
The volume of water per minute from B pipe = 15
15 × 7 = 105
150 - 105 = 45
45 / 10 = 4.5 = Time of A
7 - 4.5 = 2.5 minutes
Problem 9
Taro's father is two years older than his mother.
The age of Taro's elder brother is 1/3 of his father's age.
The age of Taro will become 1/3 of his mother's age in five years.
Find the age difference of Taro and his elder brother.
Answer
4 years
Solution
The age of Taro's elder brother now is set to be 1.
The age of the father = 3
The age of the mother = 3 - 2
The age of Taro after 5 years = (3 - 2 + 5) × 1/3 = 1 + 1
Taro's current age = 1 + 1 - 5 = 1 - 4 years
Problem 10
The character P is written to one face of a certain rectangular prism.
(2)
Problem 11
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.
Answer
(2) 756 cm2
(1)
Solution
(1)
Based on the development view of Fig.1, the sketch of the Fig.3 can be drawn and three diagonal lines are drawn in this sketch.
Then we can find three diagonal lines in the development view of Fig. 2 as well.
<Another solution>
Put alphabet on each vertex of the sketch as shown in Fig. 4 below.
Then put alphabet on each vertex of Fig.1 and Fig.2 as well as shown in Fig.5 and Fig.6.
According to Fig.6, it is found that three diagonal lines are EG, GC and EB.
Fig.7 is the same as Fig.1 and Fig.8 is rectangle ABCD.
Comparing these two figures, the difference of the surrounding length of two figures is four sides of circle marked.
Thus the length of one side is 24 / 4 = 6 cm.
As for the development views of Fig. 1 and Fig.2, the number of blue sides are eight in both figures.
The number of red sides in the Fig.2 is more than that of Fig.1 by two and the number of orange sides in the Fig.1 is more than that of Fig.2 by two.
Thus the difference of the length of red side and orange side is 10 / 2 = 5 cm and the length of the orange line = 14 - 5 = 9 cm.
Therefore the volume of this rectangular prism is 14 × 6 × 9 = 756 cm3.
Problem 11
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.
Those who did not give right answers of all three problems were 5%.
Find the number of students of this grade.
Answer
180 persons
Solution
100% - 5% - 15% = 80% (Red area in Fig.2)
80% - 18 persons = 70%
18 persons = 80% - 70% = 10%
Total = 18 / 0.1 = 180 persons.
Problem 12
The figure is a circle centering on the point O with radius of 6 cm.
Since the length of PQ in 8 seconds is 8 cm, the area of of the triangle CPQ 116 seconds after P and Q leave is 8 × 6 / 2 = 24 cm2.
Problem 13
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.
Pi is assumed to be 3.14.
(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.
Answer
(1) 21 cm2
7 seconds
(2) 24 cm2
(3) 34 seconds
7 seconds
(2) 24 cm2
(3) 34 seconds
Solution
(1)
P moves at the speed of 3 cm/s.
P moves 3 × 5 = 15 cm in 5 seconds.
As 15 - 6 × 2 = 3 cm, PO = 3 cm.
Q moves at the speed of 2 cm/s.
Q moves 2 × 5 = 10 cm in 5 seconds.
As Bo = 6 cm, QO = 10 - 6 = 4 cm.
Therefore the area of triangle CPQ = 6 × (3 + 4) / 2 = 21 cm2.
The direction in 5 seconds which P and Q moves is as in Fig. 1.
In 6 seconds, as the position of P is at O and Q is at B, PQ = 6cm.
In 7 seconds, since the position of P is 3 cm from O and Q is 2 cm from B, PQ = 7cm which is the same length as above.
Therefore the time when it becomes same area of 21 cm2 is 7 seconds.
(2)
P returns to A in 6 × 2 / 3 = 4 seconds and Q returns to B in 6 × 2 /2 = 6 seconds.
It is 12 seconds when P and Q return to the starting point at the same time.
The area after 116 seconds is the same as 8 seconds according to that 116 / 12 = 9 remainder 8.
The length of PQ in 12 seconds is as shown in the below table.
(3)
The area of the circle is 6 × 6 × 3.14 = 113.04 cm2.
The area of triangle CPQ = 113.04 - 107.04 = 6 cm2.
PQ = 6 cm2 × 2 / 6 cm = 2 cm.
The first time when PQ = 2cm is 2 seconds and 2nd time is 10 seconds according to the above table.
As the cycle is 12 seconds, the time of 6th times is 2, 10, 14, 24, 26, 34 seconds.
Problem 13
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.
Answer
(1) 3360 m
(2) 40 m/m
(2) 40 m/m
Solution
(1)
The numbers of 5 and 7 in Fig.1 below shows the speed of Taro and Hanako.
According to Fig.1, the distance of BC and AC is set to be 5 and 7 respectively as in Fig.2.
The lower part of Fig.2 shows movement of Taro and Hanko until the 2nd meeting from the 1st meeting.
As shown in this figure the distance of CD is set to be 7 - 5 = 2 which is equivalent to 560 m.
As the distance of AB is 5 + 7 = 12, it is 560 m / 2 × 12 = 3360 m.
(2)
Hanako moved BC at the speed of 7 and BD at the speed of 5.
The time ratio of BC and BD = 5 / 7 : 7 / 5 = 25 : 49.
The time for Hanako to move BD = 74 × 49/25+49 = 49 minutes.
The distance of BD = 560 / 2 × 7 = 1960 m.
Therefore the speed = 1960 / 49 = 40 m/m.
