Time : 50 minutes
Answer : End of the problem
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
There are train A at 72 km per hour and train B at 126 km per hour.
It takes 10 seconds for train A and B to pass each other after meeting.
It took 20 seconds for A and 42 seconds for B to finish passing a railway bridge after start passing respectively.
(1) How many meters is the length of the train A?
(2) How many meters is the length of the railway bridge?
Problem 2
Yuriko made a following plan to buy snacks for picnic.
① The total money for shopping is 500 yen and all money should be used without remainder.
② Should not buy more than one piece of the same kind of snack.
(1) When the kind of snacks is as shown in the table below, how many ways to buy snacks are there altogether?
(2) When the number of one kind of snack at 100 yen is increased, how many ways to buy snacks are increased?
Problem 3
There is a paper tape of width 1cm and length 30m.
It's cut into each 10 cm and connected to be squares as many as possible as shown in the figure below.
Noted that when connecting paper tapes, 1 cm of tab for sticking will be used.
(1) How many squares can be made by using this paper tape?
(2) How many cm is the total length of the tape used as a tab for sticking?
Problem 4
(1) There are a circle with diameter 12 cm and an equilateral triangle with one side of 12cm as shown in the figure below.
Find the area of the shaded portion.
Pi is assumed to be 22/7.
Problem 4
2 ) The figure below shows three squares overlap and point E is the middle point of side AD.
Area ratio of X and Y is 3 : 1.
Area ratio of X and Y is 3 : 1.
Find the area ratio of Z and square ABCD.
Problem 5
There is a container of quadratic prism whose base is a trapezoid of which the side AD and BC are parallel as shown in the figure below.
Point I is a point on the side BC and the side AI and BC are vertical.
BI = 1cm. CI = 3cm. AD = 5cm. The side BE = 35cm.
Two dashboards were put in this container vertically to the base.
One dashboard was put in to pass A and I, and another was put in to pass D and I.
One container including the side AB is set to i, the second container including the side AD is set to ii and the third container including the side CD is set to iii.
The same amount of water was put in these 3 containers.
The thickness of the dashboard isn't considered.
(1) When the same amount of water is included in containers of i, ii and iii, the height of the water surface in container ii was 6cm.
Find the height of the water surface in container ii iii when removing a dashboard between container ii and iii.
(2) Furthermore, when dashboard between container i and ii were also removed, find the height of the water surface of this quadratic prism.
(3) After (2), when a cylinder of radius 2cm and height 20cm was put in this container to the bottom vertically, find the height of the waster surface.
AI = 8cm. Pi is assumed to be 22/7.
Answer
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
There are train A at 72 km per hour and train B at 126 km per hour.
It takes 10 seconds for train A and B to pass each other after meeting.
It took 20 seconds for A and 42 seconds for B to finish passing a railway bridge after start passing respectively.
(1) How many meters is the length of the train A?
(2) How many meters is the length of the railway bridge?
Answer
(1) 345 m
(2) 495 m
Solution
(1) 72 km per hour is 20 m per second and 126 km per hour is 35 m per second.
The total length of 2 trains is (20 + 35) ×10 =550m.
20 × 42 = 840 m.
35 × 20 = 700 m.
The difference of the length of 2 trains is 840 - 700 = 140m.
The length of A is (550 + 140) / 2 = 345m.
(2) 20 × 42 - 345 = 495 m.
Problem 2
Yuriko made a following plan to buy snacks for picnic.
① The total money for shopping is 500 yen and all money should be used without remainder.
② Should not buy more than one piece of the same kind of snack.
(1) When the kind of snacks is as shown in the table below, how many ways to buy snacks are there altogether?
(2) When the number of one kind of snack at 100 yen is increased, how many ways to buy snacks are increased?
Answer
(1) 5 ways
(2) 7 ways
Solution
(1) 5 ways as shown in the table below.
(2) 7 ways as shown in the table below.
Problem 3
There is a paper tape of width 1cm and length 30m.
It's cut into each 10 cm and connected to be squares as many as possible as shown in the figure below.
Noted that when connecting paper tapes, 1 cm of tab for sticking will be used.
(1) How many squares can be made by using this paper tape?
(2) How many cm is the total length of the tape used as a tab for sticking?
Answer
(1) 99 sheets
(2) 596 cm
Solution
(1) It's possible to think one square is made of 1 sheet and 1 set of 3 sheets in the figure below.
The 2nd square is made by combining the 2nd set.
10 cm paper tape is made 3000 cm/10 cm = 300 sheets altogether.
The number of squares can be made is (300 - 1) /3 = 99 sheets.
(2) There are 2 tabs for sticking in first sheet as shown in the figure below.
There are six tabs for 1 set from the following set.
Therefore total length of the tape used for tabs for sticking is 2 cm + 6 cm × 99 = 596 cm.
Problem 4
(1) There are a circle with diameter 12 cm and an equilateral triangle with one side of 12cm as shown in the figure below.
Find the area of the shaded portion.
Pi is assumed to be 22/7.
Answer
(1) 264/7 cm2
Solution
( 1 ) Shaded area can be transformed with same area as shown in the figure below .
The area to find is 6 × 6 × 22/7 × (60×2) /360 = 264/7 cm2.
Problem 4
2 ) The figure below shows three squares overlap and point E is the middle point of side AD.
Area ratio of X and Y is 3 : 1.
Area ratio of X and Y is 3 : 1.
Find the area ratio of Z and square ABCD.
Answer
(2) 1：96
Solution
Because X : Y = 3 : 1, the area in square OFCG is shown as in the figure below.
Because the area of square ABCD will be (1 + 3 + 9 + 11) × 4 = 96, the area ratio of Z and square ABCD is 1 : 96.
Problem 5
There is a container of quadratic prism whose base is a trapezoid of which the side AD and BC are parallel as shown in the figure below.
Point I is a point on the side BC and the side AI and BC are vertical.
BI = 1cm. CI = 3cm. AD = 5cm. The side BE = 35cm.
Two dashboards were put in this container vertically to the base.
One dashboard was put in to pass A and I, and another was put in to pass D and I.
One container including the side AB is set to i, the second container including the side AD is set to ii and the third container including the side CD is set to iii.
The same amount of water was put in these 3 containers.
The thickness of the dashboard isn't considered.
(1) When the same amount of water is included in containers of i, ii and iii, the height of the water surface in container ii was 6cm.
Find the height of the water surface in container ii iii when removing a dashboard between container ii and iii.
(2) Furthermore, when dashboard between container i and ii were also removed, find the height of the water surface of this quadratic prism.
(3) After (2), when a cylinder of radius 2cm and height 20cm was put in this container to the bottom vertically, find the height of the waster surface.
AI = 8cm. Pi is assumed to be 22/7.
Answer
(1) 7.5 cm
(2) 10 cm
(3) 630/41 cm
Solution
(1) Because BI = 1 cm, AD = 5cm, IC = 3 cm and the height of the water surface in ii is 6 cm, the volume of the water put in ii is set to be 5 × 6 = 30.
The height of the water surface to be found is (30 + 30)/(5+3) = 7.5 cm.
(2) 30 × 3 / (1 + 5 + 3) = 10 cm.
(3) The area of trapezoid ABCD is (5 + 4) × 8 × 1/2 = 36 cm2.
The base area of the cylinder is 2 × 2 × 22/7 = 88/7 cm2.
Because 36 - 88/7 = 164/7 and 36 : 164/7 = 63 : 41, it's 10 cm × 63/41 = 630/41 cm.
