Time : 40 minutes
Passing mark : 70%
Problem 1
Problem 2
Problem 4
The ratio of the number of candies which three persons, Taro, Jiro, and Hanako had was 5 : 4 : 2.
Jiro got 14 candies from Taro and ate four.
After that, Jiro gave the half of candies he had to Hanako.
Then the number of candies which Taro and Hanako became equal.
Find the number of candies Taro had first.
Problem 5
There are in total 400 balls of 4 colors of red, yellow, blue and green.
Problem 6
Problem 7
Problem 8
Problem 9
Problem 1
Problem 2
Problem 4
The ratio of the number of candies which three persons, Taro, Jiro, and Hanako had was 5 : 4 : 2.
Jiro got 14 candies from Taro and ate four.
After that, Jiro gave the half of candies he had to Hanako.
Then the number of candies which Taro and Hanako became equal.
Find the number of candies Taro had first.
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Passing mark : 70%
Problem 1
Calculate the followings.
(1)
7 × 5 - 238 / 17
(2)
5/4 × ( 5/6 - 4/9 ) / 7/12 + 1/2
(3)
1/2 + ( 0.8 + 1/4 ) × (2 - 12/7) - {( 3/5 + 0.3 ) / 3/2 - 1/5}
Problem 2
(1)
When 6/5 is divided by X and add 1/4, the quotient is 2.5.
Find X.
(2)
Find the number of numbers to be divisible by 3 or 5 in integers from 1 to 200.
(3)
When eight pencils are distributed to each student, 27 pencils run short. When five pencils are distributed to each student, 36 pencils remain. Find a number of students.
(4)
When I run the distance where it takes 14 minutes to run at the speed of 18 km/h, at the speed of 120 m/m, find the time for me to take.
(5)
How many ways of method of putting two oranges and two apples on three plates of white, green and blue are there?
Noted that there shall be no empty plate.
Problem 3
The figure below is a graphic drawn with semicircles with radii 2 cm and 4 cm.
Find the area of the portion of a shadow.
Pi is assumed to be 3.14.
Problem 4
Jiro got 14 candies from Taro and ate four.
After that, Jiro gave the half of candies he had to Hanako.
Then the number of candies which Taro and Hanako became equal.
Find the number of candies Taro had first.
Problem 5
There are in total 400 balls of 4 colors of red, yellow, blue and green.
We threw balls to a box.
The number of balls which were not in the box was 100 or less pieces.
Among the balls in a box, 1/4 was red, 1/6 was yellow and 3/7 was blue.
Find the number of green balls in the box.
Problem 6
There are ten cards.
One of odd numbers from 1 to 19 is written on each card.
When taking out two cards in order four times, each sum was 16, 18, 24, and 34 in order.
Find the numbers of the 2nd set and the 3rd set, respectively.
Problem 7
One day when Taro left home toward the school, 7 minutes after his leaving his mother noticed the thing left behind of Taro.
Although she pursued at the speed of 80 m/m immediately, she thought that she could not catch up Taro.
Then she borrowed a bicycle at the house of Taro's friend which is on the way and kept pursuing.
Although she took 3 minutes to borrow, she caught up with Taro just in front of the school gate.
The way from the house to a school is straight and the distance is 1500 m.
The graph below expresses the relation between the time after Taro leaves home and the distance of two persons.
Answer the following questions.
Noted that the speed which Taro walks, the speed which his mother walks, and the speed of the bicycle are constant respectively.
(1) Find the speed which Taro walks at the speed per minute.
(2) Find the distance from the house of Taro to his friend's house.
(3) Find the speed of the bicycle at the speed per minute.
Problem 8
The vessel below is 10 cm in height and the bottom is the form which combined a rectangle and semicircles as shown in Fig. 1.
Putting water into this vessel full and with attaching the portions of A and B to a floor, 45 degrees was leaned and water was spilt.
Fig. 2 is a figure showing that it was seen from the side.
Find the amount of the remaining water.
Pi is assumed to be 3.14.
Problem 9
As shown in a lower figure, the integer was clockwise arranged in order from 1.
The located integer will be expressed with which direction of what round it is.
For example, 30 is the number of the northeast of the 3rd round.
Answer the following questions.
(1) With which direction of what round is 64 located ?
(2) Find the number of the northwest of the 10th round.
(3) It was 4160 when numbers of the northwest, northeast, southeast, and southwest of a certain round was added.
Find the number of the southeast of this round.
Answer / Solution
Problem 1
Calculate the followings.
(1)
7 × 5 - 238 / 17
(2)
5/4 × ( 5/6 - 4/9 ) / 7/12 + 1/2
(3)
1/2 + ( 0.8 + 1/4 ) × (2 - 12/7) - {( 3/5 + 0.3 ) / 3/2 - 1/5}
Answer
(1) 21
(2) 4/3
(3) 2/5
Problem 2
(1)
When 6/5 is divided by X and add 1/4, the quotient is 2.5.
Find X.
Answer
8/15
Solution
6/5 / X + 1/4 = 2.5
X = 6/5 / (2.5 - 1/4) = 8/15
(2)
Find the number of figures to be divisible by 3 or 5 in integers from 1 to 200.
Answer
93
Solution
200 / 3 = 68 remainder 2
200 / 5 = 40
200 / 15 = 13 remainder 5
66 + 40 - 13 = 93
(3)
When eight pencils are distributed to each student, 27 pencils run short. When five pencils are distributed to each student, 36 pencils remain. Find a number of students.
Answer
21 persons
Solution
27 + 36 = 63
8 - 5 = 3
63 / 3 = 21 persons
(4)
When I run the distance where it takes 14 minutes to run at the speed of 18 km/h, at the speed of 120 m/m, find the time for me to take.
Answer
35 minutes
Solution
18 × 14/60 = 4.2 km
4.2 × 1000 = 4200 m
4200 / 120 = 35 minutes
(5)
How many ways of method of putting two oranges and two apples on three plates of white, green and blue are there?
Noted that there shall be no empty plate.
Answer
12 ways
Solution
① (OO, A, A) = 3 ways
② (AA, O, O) = 3 ways
③ (OA, O, A) = 3 × 2 × 1 = 6 ways
3 + 3 + 6 = 12 ways
Problem 3
The figure below is a graphic drawn with semicircles with radii 2 cm and 4 cm.
Find the area of the portion of a shadow.
Pi is assumed to be 3.14.
Answer
67.4 cm2
Solution
4 × (2 + 8) - 2 × 2 + 4 × 4 × 3.14 × 1/2 + 4 × 4 × 3.14 × 1/4 - 2 × 2 × 3.14 × 1/2
= 40 - 4 + (8 + 2) × 3.14 = 36 + 31.4 =67.4 cm2
Problem 4
Jiro got 14 candies from Taro and ate four.
After that, Jiro gave the half of candies he had to Hanako.
Then the number of candies which Taro and Hanako became equal.
Find the number of candies Taro had first.
Answer
95 pieces
Solution
Taro = 5 pieces, Jiro = 4 pieces, Hanko = 2 pieces
Taro : 5 - 14
Jiro : 4 + 14 - 4 = 4 + 10
Hanako : 2 + (2 + 5) = 4 + 5
Taro = Hanako, 5 -14 = 4 + 5
5 -4 = 5 + 14
1 = 19
Taro = 5 = 19 × 5 = 95 pieces
Problem 5
There are in total 400 balls of 4 colors of red, yellow, blue and green.
We threw balls to a box.
The number of balls which were not in the box was 100 or less pieces.
Among the balls in a box, 1/4 was red, 1/6 was yellow and 3/7 was blue.
Find the number of green balls in the box.
Answer
52 pieces
Solution
Number of balls in the box = 400 - 100 = 300. Then 300 < (balls in the box) < 400.
Ratio of green in the box = 1 - (1/4 - 1/6 - 3/7) = 13/84
Number of ball is multiple of 84.
84 × 3 = 252, 84 × 4 =336, 84 × 5 = 420
It is 336.
336 × 13/84 = 52 pieces
Problem 6
There are ten cards.
One of odd numbers from 1 to 19 is written on each card.
When taking out two cards in order four times, each sum was 16, 18, 24, and 34 in order.
Find the numbers of the 2nd set and the 3rd set, respectively.
Answer
The 2nd set : 1 and 17,
The 3rd set : 11 and 13
Solution
4th set = 34 ＝15＋19
1st = (3,13), (5,11), (7,9)
2nd = (1,17), (5,13), (7,11)
3rd = (7,17), (11,13)
Above all, possible combination is 2nd(1,17) and 3rd(11,13).
Problem 7
One day when Taro left home toward the school, 7 minutes after his leaving his mother noticed the thing left behind of Taro.
Although she pursued at the speed of 80 m/m immediately, she thought that she could not catch up Taro.
Then she borrowed a bicycle at the house of Taro's friend which is on the way and kept pursuing.
Although she took 3 minutes to borrow, she caught up with Taro just in front of the school gate.
The way from the house to a school is straight and the distance is 1500 m.
The graph below expresses the relation between the time after Taro leaves home and the distance of two persons.
Answer the following questions.
Noted that the speed which Taro walks, the speed which his mother walks, and the speed of the bicycle are constant respectively.
(1) Find the speed which Taro walks at the speed per minute.
(2) Find the distance from the house of Taro to his friend's house.
(3) Find the speed of the bicycle at the speed per minute.
Answer
(1) 75 m/m
(2) 480 m
(3) 255 m/m
Solution
(1) 1500 / 20 = 75 m/m
(2) 75 × 7 = 525
525 - 495 = 30 m
30 / (80-75) = 6 minutes
80 × 6 = 480 m
(3) (495 + 75 3) / ( X - 75) = 4 minutes
X = 180
180 + 75 = 255 m/m
Problem 8
The vessel below is 10 cm in height and the bottom is the form which combined a rectangle and semicircles as shown in Fig. 1.
Putting water into this vessel full and with attaching the portions of A and B to a floor, 45 degrees was leaned and water was spilt.
Fig. 2 is a figure showing that it was seen from the side.
Find the amount of the remaining water.
Pi is assumed to be 3.14.
Answer
292.48 cm3
Solution
24 + 2 × 2 × 3.14 = 36.56 cm2
36.56 × 4 / 2 = 73.12 cm3
36.56 × 10 - 73.12 = 292.48 cm3
Problem 9
As shown in a lower figure, the integer was clockwise arranged in order from 1.
The located integer will be expressed with which direction of what round it is.
For example, 30 is the number of the northeast of the 3rd round.
Answer the following questions.
(1) With which direction of what round is 64 located ?
(2) Find the number of the northwest of the 10th round.
(3) It was 4160 when numbers of the northwest, northeast, southeast, and southwest of a certain round was added.
Find the number of the southeast of this round.
Answer
(1) the number of the southeast of fourth round
(2) 440
(3) 1024
Solution
(1) 1st round = 4 = 2 × 2
2nd round = 16 = 4 × 4
3rd round = 36 = 6 × 6
4th round = 64 = 8 × 8
(2) 1st round = 1 = 1 × 1
2nd round = 9 = 3 × 3
3rd round = 25 = 5 × 5
Then 11th round = 21 × 21 = 441
The number of northwest = 441 - 1 =440
(3) Sum of 1st round = 2 + 4 + 6 + 8 = 20 = 5 × 4 = 5 × 4 × 1
2nd round = 12 + 16 + 20 + 24 = 72 = 9 × 8 = 9 × 4 × 2
3rd round = 30 + 36 + 42 + 48 = 156 = 13 × 12 = 13 × 4 × 3
4160 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 13 = 65 × 64 = 65 × 4 × 16
4160 is sum of 16th round.
16 × 2 = 32
32 × 32 = 1024
