Answer : End of the Problem
Problem 1
Problem 2
(3) Train A at a speed of 1,200m per minute has begun to enter a certain tunnel.
(4) There is a rectangle A whose ratio between length and width is 3 : 5.
Problem 4
Problem 5
(1) Y is equal to X and point P rebounds at the position of 30 cm from vertex B.
(1) Calculation
12 + (6 × 8 - 18 / 3) / 3 =
(2) Calculation
13/6 - 4/3 / (1 - 0.2) × 1/4 =
(3) Calculation
(3/5 - 0.25 + 7/12 - 13/30) × 60 =
(4) Find X.
435 / (12 + 9 × X) = 24 remainder 3
(5) Find Y.
(2 × Y + 1) / (3 × Y - 2) = 3/4
Problem 2
(1) Taro began to read a certain book by 7 pages a day and Jiro began by 5 pages a day.
They started reading at the same time.
On the day when Taro finished reading, there still 114 pages were left for Jiro.
Find the pages of this book.
(2) When a consumption tax of 5% is added to the price of a certain goods, it'll be 2014 yen.
(3) When the number of tens digit of double digits integer is subtracted from the number of ones digit, it becomes five.
(4) The figure below is a graphic that is a combination of square and sector.
(5) The figure below is a figure overlapping three rectangular papers with 3cm in length and 15cm in width.
Problem 3
(2) Mother's age is 5 times of the daughter's age now.
(2) When a consumption tax of 5% is added to the price of a certain goods, it'll be 2014 yen.
Noted that less than 1 yen of the amount of the consumption tax is cut off.
Find the price of this book.
(3) When the number of tens digit of double digits integer is subtracted from the number of ones digit, it becomes five.
As for the number that replaced the number of tens digit and ones digit, it becomes bigger by seven than the double of the original integer.
Find this integer.
(4) The figure below is a graphic that is a combination of square and sector.
Find the difference of the area of A and B.
Pi is assumed to be 3.14.
(5) The figure below is a figure overlapping three rectangular papers with 3cm in length and 15cm in width.
The area of the shadow is 102 cm2.
Find the total length of bold lines.
Problem 3
(1) 40 students played a game.
The score was either two points, four points, six points or ten points.
There were four people who had two points and eight people who had ten points.
The average score was 5.8 points.
Find the number of people who had four points and six points.
(2) Mother's age is 5 times of the daughter's age now.
Four years later, mother's age becomes less 4 years less than 4 times of the daughter's age.
Find the the daughter's age now.
In addition, how many years later from now does the mother's age become 4 years less than 3 times of the daughter's age?
(3) Train A at a speed of 1,200m per minute has begun to enter a certain tunnel.
Train B of 1,500m per minute has begun to enter at the opposite side eight seconds later.
The top of train A and B came across in the middle of the tunnel.
Train B completely left the tunnel one minute 17 seconds later after train A began to enter the tunnel.
Find the length of this tunnel and the length of train B.
(4) There is a rectangle A whose ratio between length and width is 3 : 5.
The length of this rectangle A is lengthened by 6cm and the width is shortened by 4 cm to be a rectangle B.
The ratio between length and width of the rectangle B became 9 : 8.
Find the vertical length rectangle A.
Moreover, find an area ratio of rectangle A and B with the least integer.
Problem 4
In order to go from a house to school, Hanako walks from the house to the bus stop first.
She takes a bus from there and walks again after she gets off the bus.
For a while after Hanako left the house, her mother noticed s thing left behind and ran after the same way by car.
Then both of them have arrived at school at the same time.
Below is the graph which showed time and distance between Hanako and her mother after Hanako left the house.
Speed Hanako walks and speed of the bus and the car are fixed respectively.
She did not wait for a bus at a bus stop.
Answer the following questions.
(1) Find the speed per minute Hanako walks and the speed per minute of the bus respectively.
(2) Find the distance from the house to the school.
(3) Find the speed per hour of the car.
Problem 5
Point P leaves vertex A and moves straight the inside of the square of one side of 80cm according to the next rule.
< Rule >
・ When P hits each side, it rebounds.
・ When P hits each vertex, it stops.
・ When P hits each side, it rebounds.
・ When P hits each vertex, it stops.
As shown in the Fig.1 below, the angle point P hits is set to X and the angle P rebounds is set to Y.
Answer to the following questions.
(1) Y is equal to X and point P rebounds at the position of 30 cm from vertex B.
Which is the vertex that is the nearest to the position where P rebounds at the fourth time?
In addition, find the distance from the vertex.
(2) In the case of (1), how many times does P rebound before it stops?
(3) Y is half of X. The locus before P rebounds at fourth time became as shown in the Fig.2 below.
Find the angle of Z.
Problem 1
Problem 2
(3) Train A at a speed of 1,200m per minute has begun to enter a certain tunnel.
(1) Calculation
12 + (6 × 8 - 18 / 3) / 3 =
Answer
26
(2) Calculation
13/6 - 4/3 / (1 - 0.2) × 1/4 =
Answer
7/4
(3) Calculation
(3/5 - 0.25 + 7/12 - 13/30) × 60 =
Answer
30
(4) Find X.
435 / (12 + 9 × X) = 24 remainder 3
Answer
2/3
(5) Find Y.
(2 × Y + 1) / (3 × Y - 2) = 3/4
Answer
10
Problem 2
(1) Taro began to read a certain book by 7 pages a day and Jiro began by 5 pages a day.
They started reading at the same time.
On the day when Taro finished reading, there still 114 pages were left for Jiro.
Find the pages of this book.
(2) When a consumption tax of 5% is added to the price of a certain goods, it'll be 2014 yen.
(3) When the number of tens digit of double digits integer is subtracted from the number of ones digit, it becomes five.
(4) The figure below is a graphic that is a combination of square and sector.
(5) The figure below is a figure overlapping three rectangular papers with 3cm in length and 15cm in width.
Problem 3
(2) Mother's age is 5 times of the daughter's age now.
Answer
399 pages
Solution
The difference of pages a day is 7 - 5 = 2 pages.
As the difference became 114 pages, number of days they read is 114 / 2 = 57 days.
Therefore the number of pages of this book is 7 × 57 = 399 pages.
(2) When a consumption tax of 5% is added to the price of a certain goods, it'll be 2014 yen.
Noted that less than 1 yen of the amount of the consumption tax is cut off.
Find the price of this book.
Answer
1919 yen
Solution
2014/ (1 + 0.05) is =1918.0--.
Therefore the price of this book is 1918 + 1 = of 1919 yen.
(3) When the number of tens digit of double digits integer is subtracted from the number of ones digit, it becomes five.
As for the number that replaced the number of tens digit and ones digit, it becomes bigger by seven than the double of the original integer.
Find this integer.
Answer
38
Solution
This integer is assumed to be AB. B - A = 5---- ①
10 × B + A = (10 × A+ B) × 2 + 7 then it is to be 8 × B - 19 × A = 7-----②
① is multiplied by 19 to be 19 × B - 19 × A = 95.
According to ① and ②, 11 × B = 95 - 7 = 88.
Therefore, B = 88 / 11 = 8. A = 8 - 5 = 3.
(4) The figure below is a graphic that is a combination of square and sector.
Find the difference of the area of A and B.
Pi is assumed to be 3.14.
Answer
0.56 cm2
Solution
A = 4 × 4 × 3.14 × 45/360 – 4 × 4 × 1/2 × 1/2 = 6.28 – 4 = 2.28 cm2
B = 4 × 4 × 1/2 - 4 × 4 × 3.14 × 45/360 = 8 – 6.28 = 1.72 cm2
A – B = 2.28 – 1.72 = 0.56 cm2
B = 4 × 4 × 1/2 - 4 × 4 × 3.14 × 45/360 = 8 – 6.28 = 1.72 cm2
A – B = 2.28 – 1.72 = 0.56 cm2
(5) The figure below is a figure overlapping three rectangular papers with 3cm in length and 15cm in width.
The area of the shadow is 102 cm2.
Find the total length of bold lines.
Answer
64 cm
Solution
The area of one rectangle is 3 × 15 = 45cm2.
Because there are three sheets, it is 45 × 3 = 135cm2 in total.
The difference with the shadow area is 135 - 102 = 33cm2.
This difference is the area of the part that rectangles overlapping.
Each three figures overlapped are lozenges.
The total length of one side of three lozenges is 33/3=11cm.
The total length of perimeter of three rectangles is (3 + 15 ) × 2 × 3 = 108cm.
The total length of perimeter of three lozenges is 11 × 4 = 44cm.
Therefore, the total length of the bold lines is 108 - 44 = 64cm.
Problem 3
(1) 40 students played a game.
The score was either two points, four points, six points or ten points.
There were four people who had two points and eight people who had ten points.
The average score was 5.8 points.
Find the number of people who had four points and six points.
Answer
Four points : 12 persons
Six points : 16 persons
Solution
The total score is 5.8 × 40 = 232.
Total score of 2 points and 10 points is 2 × 4 + 10 × 8 = 88.
Total score of 4 points and 6 points is 232 - 88 = 144.
Total number of people of 4 points and 6 points are 40 - (4 + 8) = 28.
Therefore number of people of 6 points is (144 - 4 × 28)/(6 - 4) = 16.
Number of people of 4 points is 28 - 16 = 12.
(2) Mother's age is 5 times of the daughter's age now.
Four years later, mother's age becomes less 4 years less than 4 times of the daughter's age.
Find the the daughter's age now.
In addition, how many years later from now does the mother's age become 4 years less than 3 times of the daughter's age?
Answer
8 years old,
10 years later
Solution
When the current age of the daughter is set to X years old, the age of the mother is X × 5.
Their ages 4 years later are expressed as X × 5 + 4 = (X + 4) × 4 - 4.
When this formula is calculated, X = 8.
Mother is 8 × 5 = 40 years old now.
According to the formula of 40 + Y = (8 + Y) × 3 - 4, Y = 10 years later.
(3) Train A at a speed of 1,200m per minute has begun to enter a certain tunnel.
Train B of 1,500m per minute has begun to enter at the opposite side eight seconds later.
The top of train A and B came across in the middle of the tunnel.
Train B completely left the tunnel one minute 17 seconds later after train A began to enter the tunnel.
Find the length of this tunnel and the length of train B.
Answer
1600 m,
125 m
(4) There is a rectangle A whose ratio between length and width is 3 : 5.
Answer
Solution
The speed per second of A and B is 1200/60 = 20 m/s and 1500/60 = 25 m/s.
It takes 160/(25 - 20) = 32 seconds for A and B to be the difference of 20 × 8 = 160m.
Therefore, the length of the tunnel is 25 × 32 × 2 = 1,600m.
Because B went along the tunnel in 1 minute 9 seconds - 8 seconds = 1 minute 9 seconds = 69 seconds, the length of B is 25 × 69 - 1600 = 125m.
(4) There is a rectangle A whose ratio between length and width is 3 : 5.
The length of this rectangle A is lengthened by 6cm and the width is shortened by 4 cm to be a rectangle B.
The ratio between length and width of the rectangle B became 9 : 8.
Find the vertical length rectangle A.
Moreover, find an area ratio of rectangle A and B with the least integer.
12 cm,
5 : 6
Problem 4
Problem 5
(1) Y is equal to X and point P rebounds at the position of 30 cm from vertex B.
Solution
(③ + 6) : (⑤ - 4) = 9 : 8.
Thus (⑤ - 4) × 9 = (③ + 6) × 8.
① = 84/21 = 4.
The length is 4 × 3 = 12cm and the width is 4 × 5 = 20cm.
12 × 20 : (12 + 6) × (20 - 4) = 240 : 288 = 5 : 6.
Problem 4
In order to go from a house to school, Hanako walks from the house to the bus stop first.
She takes a bus from there and walks again after she gets off the bus.
For a while after Hanako left the house, her mother noticed s thing left behind and ran after the same way by car.
Then both of them have arrived at school at the same time.
Below is the graph which showed time and distance between Hanako and her mother after Hanako left the house.
Speed Hanako walks and speed of the bus and the car are fixed respectively.
She did not wait for a bus at a bus stop.
Answer the following questions.
(1) Find the speed per minute Hanako walks and the speed per minute of the bus respectively.
(2) Find the distance from the house to the school.
(3) Find the speed per hour of the car.
Answer
(1) 70 m/m, 560 m/m
(2) 7980 m
(3) 39.9 km/h
Solution
(1) 490/7=70 m/m (2730 - 490) / (11 - 7) = 560 m/m
(2) 490 + 560 × (20 - 7) + 70 × (23 - 20) = 490 + 7280 + 210 = 7980 m.
(3) The speed per minute of the car is assumed X m/m.
As (X - 560) × (20 - 11) + (X - 70) × (23 - 20) = 2730, X × 12 = 2730 + 5040 + 210 = 7980.
Therefore X = 7980/12 = 665 m/m.
The speed per hour is 665 × 60/1000 = 39.9 km/h.
Problem 5
Point P leaves vertex A and moves straight the inside of the square of one side of 80cm according to the next rule.
< Rule >
・ When P hits each side, it rebounds.
・ When P hits each vertex, it stops.
・ When P hits each side, it rebounds.
・ When P hits each vertex, it stops.
As shown in the Fig.1 below, the angle point P hits is set to X and the angle P rebounds is set to Y.
Answer to the following questions.
(1) Y is equal to X and point P rebounds at the position of 30 cm from vertex B.
Which is the vertex that is the nearest to the position where P rebounds at the fourth time?
In addition, find the distance from the vertex.
(2) In the case of (1), how many times does P rebound before it stops?
(3) Y is half of X. The locus before P rebounds at fourth time became as shown in the Fig.2 below.
Find the angle of Z.
Answer
(1) C, 10 cm
(2) 9 times
(3) 64 degrees
Solution
(1) P rebounds as shown in Fig.3.
The position rebounds at the fourth time is P4 and the near vertex is C.
Because P1P4 = AP2 = 60cm, CP4 = (30 + 60) - 80 = 10 cm.
Because P1P4 = AP2 = 60cm, CP4 = (30 + 60) - 80 = 10 cm.
(2) According to 60 × 4 = 80 × 3, the locus is as shown in Fig.4.
It is nine times in all.
(3) When determining angles from Q to W as shown in Fig.5 below, ∠ R= ∠ Z/2,
∠ S =90 ° - Z/2
∠ T = ∠ S/2 = 45 ° - Z/4
∠ U = 90 ° - ∠ T = 45 ° + Z/4
∠ V = ∠ U/2 = 22.5 ° + Z/8
∠ W = ∠ Z
According to ∠ W + ∠V + 85.5 ° = 180°,
Z + 22.5 + Z/8 + 85.5 = 180, then
Z + Z/8 = 72
Therefore Z = 64 °
∠ U = 90 ° - ∠ T = 45 ° + Z/4
∠ V = ∠ U/2 = 22.5 ° + Z/8
∠ W = ∠ Z
According to ∠ W + ∠V + 85.5 ° = 180°,
Z + 22.5 + Z/8 + 85.5 = 180, then
Z + Z/8 = 72
Therefore Z = 64 °
