Time : 50 minutes
Answer : End of problem
Problem 1
(4) When a certain product is sold at a regular price, the profit is 45 yen per one piece.
(5) When I divided a certain number by 123, a quotient was 23 and remainder was 98 .
(6) There is a semicircular O with radius 6 cm as shown in the figure below.
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
(1) Calculation
(1/2 - 1/3 + 1/5) / (1/4 + 1/6 + 2/15) =
(2) Find X.
(4 × 6/5 - 2 / 10/3) × X + 2/5 = 2
(3) Total price of 1 pencil and 2 notebooks is 420 yen.
Total price of 4 pencils and 3 notebooks is 930 yen.
Find the price of 1 pencil and 1 notebook respectively.
(4) When a certain product is sold at a regular price, the profit is 45 yen per one piece.
The profit when selling 8 pieces at the price of 15 percent discount of the regular price and the profit when selling 12 pieces at the price of reducing 35 yen from the regular price are same.
Find the regular of this product.
(5) When I divided a certain number by 123, a quotient was 23 and remainder was 98 .
Find a quotient and a remainder when I divide this number by 41.
(6) There is a semicircular O with radius 6 cm as shown in the figure below.
The circumference of the semicircular is divided into 6 equal parts.
(6)-1 Find the area of triangle OAB.
(6)-2 Find the area of the shaded portion.
Problem 2
A figure below is the figure which overlap 2 sheets of square papers with same size.
(1) In case of the area ratio of P and Q is P : Q = 3 : 2, find the ratio of the length of CE and DE.
(2) In case of the area ratio of P and Q is P : Q = 2 : 1, find the length ratio of the circumference of P and Q.
(3) In case of the length ratio of the circumference of P and Q is 5 : 3, find the area ratio of P and Q.
Problem 3
Taro moves on foot and Jiro by bicycle on the road from A point to B point where is 16 km away from A point.
The speed Taro walks is 80 m per minute and the speed of the bicycle of Jiro is 200 m per minute.
Jiro left 30 minutes later after Taro.
Jiro passed Taro on the way and he rested for a while after he moved.
When 10 minutes passed since he began to rest, he met Taro, so he began to move by the same speed again immediately.
Taro did not take a rest and Jiro took a rest once to arrive at B point.
(1) How many hours and minutes later did Taro arrive at B point after Jiro arrived?
(2) How many km from A point did Jiro catch up with Taro?
(3) How many minutes and seconds did Jiro moved until the time when he took a rest after passing Taro?
Problem 4
A cube whose length of one side is 6 cm is put on the flat floor as shown in the figure.
This cube is rolled to the direction of the arrow with putting the side of FG on the floor so as not to be slipped until the face BFGC reaches on the floor.
(1) Find the area of the part where the line segment AF moved.
(2) Find the volume of the part where this cube moved.
Problem 5
As shown in the figure, five circles cross and have nine parts.
Number from 1 to 9 are in the nine parts one by one.
The sum of numbers in each circle becomes all 11.
(1) Find the sum of four numbers in P, Q, R and S.
(2) Find numbers in nine parts.
Noted that the number in Q is smaller than the number in R.
Problem 6
There are circle with radius 2cm centered on point O and square ABCD whose length of one side is 6cm as shown in the figure.
Point P moves inside and on the circumference of the circle.
(1) When point Q moves on the circumference of the square, find the length of the circumference of the figure which is made by the middle point of the line segment OQ moves.
(2) Find the area of the figure made by the middle point of the line segment AP moves.
(3) When point Q moves on the circumference of the square, find the area of the figure made by the middle point of the line segment PQ moves.
Answer
Problem 1
(4) When a certain product is sold at a regular price, the profit is 45 yen per one piece.
(5) When I divided a certain number by 123, a quotient was 23 and remainder was 98 .
(6) There is a semicircular O with radius 6 cm as shown in the figure below.
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
(1) Calculation
(1/2 - 1/3 + 1/5) / (1/4 + 1/6 + 2/15) =
Answer
2/3
(2) Find X.
(4 × 6/5 - 2 / 10/3) × X + 2/5 = 2
Answer
8/21
(3) Total price of 1 pencil and 2 notebooks is 420 yen.
Total price of 4 pencils and 3 notebooks is 930 yen.
Find the price of 1 pencil and 1 notebook respectively.
Answer
Pencil : 120 yen,
Notebook : 150 yen
Solution
Total price of 4 pencils and 8 notebooks is 420 × 4 = 1680 yen.
1680 - 930 = 750 yen is price of 5 notebooks.
Thus 1 notebook is 750 / 5 = 150 yen.
1 pencil is 420 - 150 × 2 = 120 yen.
(4) When a certain product is sold at a regular price, the profit is 45 yen per one piece.
The profit when selling 8 pieces at the price of 15 percent discount of the regular price and the profit when selling 12 pieces at the price of reducing 35 yen from the regular price are same.
Find the regular of this product.
Answer
200 yen
Solution
In case reducing 35 yen from the regular price, the profit is to be 45 - 35 = 10 yen per product.
10 × 12 = 120 yen which is total profit in case selling 8 pieces at the price of 15 percent discount.
The profit per product is 120 / 8 = 15 yen. 45 - 15 = 30 yen is equivalent to 15% of the regular price.
Therefor the regular price is 30 / 0.15 = 200 yen.
(5) When I divided a certain number by 123, a quotient was 23 and remainder was 98 .
Find a quotient and a remainder when I divide this number by 41.
Answer
Quotient : 71,
Remainder : 16
Solution
A certain number is 123 × 23 + 98 = 2927.
2927 / 41 = 71 with remainder 16.
(6) There is a semicircular O with radius 6 cm as shown in the figure below.
The circumference of the semicircular is divided into 6 equal parts.
(6)-1 Find the area of triangle OAB.
(6)-2 Find the area of the shaded portion.
Answer
(6)-1 6 cm2
(6)-2 33.84 cm2
Solution
(6)-1 Since ∠AOB = 30 degrees as shown in the figure below, the height of △OAB = 6 cm × 1/2 = 3 cm.
Therefore the area of △AOB = 4 × 3 × 1/2 = 6 cm2.
(6)-2 The area of shaded portion = △OCD + Sector OAC + △OAB = 3 × 6 × 1/2 + 6 × 6 × 3.14 × 60/360 + 6 = 9 + 18.84 + 6 = 33.84 cm2.
Problem 2
A figure below is the figure which overlap 2 sheets of square papers with same size.
(1) In case of the area ratio of P and Q is P : Q = 3 : 2, find the ratio of the length of CE and DE.
(2) In case of the area ratio of P and Q is P : Q = 2 : 1, find the length ratio of the circumference of P and Q.
(3) In case of the length ratio of the circumference of P and Q is 5 : 3, find the area ratio of P and Q.
Answer
(1) 2 : 3
(2) 3 : 2
(3) 4 : 1
Solution
(1) Assuming the length of AB is 1, the area of Q = 1 × CE × 1/2 × 2 = CE.
CE = 1 × 1 × 2/(2+3) = 2/5 and DE = 1 - 2/5 = 3/5.
Therefore CE : DE = 2 : 3.
(2) In this case, CE = 1 × 1/(1+2) = 1/3.
Therefore the length ratio of circumference of P and Q is (1 × 4) : (1 × 2 + 1/3 × 2) = 4 : 8/3 = 3 : 2.
(3) When the length of the circumference of P is set to be 5, the length of one side of the square is to be 5/4.
In this case the length of CE = (3 - 5/4 × 2) × 1/2 = 1/4.
Thus the area of Q = 5/4 × 1/4 × 1/2 × 2 = 5/16.
The area of P = 5/4 × 5/4 - 5/16 = 20/16.
Therefore the area ratio of P and Q = 20/16 : 5 /16 = 4 : 1.
Problem 3
Taro moves on foot and Jiro by bicycle on the road from A point to B point where is 16 km away from A point.
The speed Taro walks is 80 m per minute and the speed of the bicycle of Jiro is 200 m per minute.
Jiro left 30 minutes later after Taro.
Jiro passed Taro on the way and he rested for a while after he moved.
When 10 minutes passed since he began to rest, he met Taro, so he began to move by the same speed again immediately.
Taro did not take a rest and Jiro took a rest once to arrive at B point.
(1) How many hours and minutes later did Taro arrive at B point after Jiro arrived?
(2) How many km from A point did Jiro catch up with Taro?
(3) How many minutes and seconds did Jiro moved until the time when he took a rest after passing Taro?
Answer
(1) 1 hour 20 minutes
(2) 4 km
(3) 6 minutes 40 seconds
Solution
(1) Taro took 16000 / 80 = 200 minutes from A to B. Jiro took 16000 / 200 + 10 = 90 minutes.
Since Jiro left 30 minutes after Taro, time to be found is 200 - 90 - 30 = 80 minutes = 1 hour 20 minutes.
(2) 30 minutes after Taro's departure, Taro is 80 × 30 = 2400 m from A.
The time for Jiro to catch up with Taro is 2400 / (200 - 80) = 20 minutes.
The distance from A = 200 × 20 = 4000 m = 4 km.
(3) 80 × 10 = 800 m is the distance between Taro and Jiro when Jiro started to take a rest.
Jiro took 800 / (200 - 80) = 20/3 minutes = 6 minutes and 40 seconds to take a rest after passing Taro.
Problem 4
A cube whose length of one side is 6 cm is put on the flat floor as shown in the figure.
This cube is rolled to the direction of the arrow with putting the side of FG on the floor so as not to be slipped until the face BFGC reaches on the floor.
(1) Find the area of the part where the line segment AF moved.
(2) Find the volume of the part where this cube moved.
Answer
(1) 56.52 cm2
(2) 555.12 cm3
Solution
(1) As shown in the figure below, shadow portion is the area AF moved.
AF × AF / 2 = 6 × 6 = 36.
AF × AF = 72.
The area AF moved = AF × AF × 3.14 × 90/360 = 72× 3.14 × 90/360 = 56.52 cm2.
(2) The area of the bottom of the part the cube moved is 56.52 + 6 × 6 = 92.52 cm2.
Therefore the volume is 92.52 × 6 = 555.12 cm3.
Problem 5
As shown in the figure, five circles cross and have nine parts.
Number from 1 to 9 are in the nine parts one by one.
The sum of numbers in each circle becomes all 11.
(1) Find the sum of four numbers in P, Q, R and S.
(2) Find numbers in nine parts.
Noted that the number in Q is smaller than the number in R.
Answer
(1) 10
(2) A 8, B 7, C 6, D 5, E 9, P 3, Q 1, R 4, S 2
Solution
(1) Total from 1 to 9 is 45.
All totals in the table below are 11 × 5 = 55.
Accordingly P + Q + R + S = 55 - 45 =10.
(2) Because 10 = 1 + 2 + 3 + 4, P, Q, R, S are numbers of 1 ~ 4.
A ~ E are numbers of 5 ~ 9.
Because 11 = 9 + 2 = 8 + 3 = 7 + 4, it's considered in case of the following 6 ways of the table.
① Q = 1, R = 4, then B = 8 which is not applicable.
② Q = 1, R = 3, then C = 7 which is not applicable.
③ Q = 1, R = 4, then B = 7, C = 6, and D = 5 which is applicable.
④ Q = 1, R = 2, then B = 7 which is not applicable.
⑤ Q = 1, R = 3, then C = 7 which is not applicable.
⑥ Q = 1, R = 2, then C = 8 which is not applicable.
According to ① ~ ⑥, only the case of ③ can be applied.
Problem 6
There are circle with radius 2cm centered on point O and square ABCD whose length of one side is 6cm as shown in the figure.
Point P moves inside and on the circumference of the circle.
(1) When point Q moves on the circumference of the square, find the length of the circumference of the figure which is made by the middle point of the line segment OQ moves.
(2) Find the area of the figure made by the middle point of the line segment AP moves.
(3) When point Q moves on the circumference of the square, find the area of the figure made by the middle point of the line segment PQ moves.
Answer
(1) 12 cm
(2) 3.14 cm2
(3) 23.14 cm2
Solution
(1) The figure made when the middle point of the line segment OQ moves is shown as a red square in the figure below.
The length of one side of this square is 1/2 of the length of one of square ABCD.
Therefore 6 cm × 1/2 × 4 = 12 cm.
(2) The figure made when the middle point of the line segment AP moves is shown as a red circle in the figure below.
The radius of this red circle is 1/2 of the radius of the circle O and it is 2 cm × 1/2 = 1 cm.
Therefore the area is 1 × 1 × 3.14 = 3.14 cm2.
(3) Considering the result of (1) and (2) as a reference, the figure made when the middle point of the line segment PQ moves is shown as a red shadow in the figure below.
The length of one side of outside red square is 1 × 2 × 2 + 1 = 5 cm.
Therefore the area is 5 × 5 - 1 × 1 - (2 × 2 - 1 × 1 × 3.14) = 23.14 cm2.
