TOHO-2014

Time : 50 minutes

Answer

Problem 1

Calculation

(1) 17/10 - 5/6 + 4/5

(2) 3.5 ÷ (2.2 - 0.8) + 0.6 × 4.5

(3) (0.2 - 1/8 × 16/15) ÷ (7/6 - 0.75)

Problem 2

(1) Length of both side of a square is made 1 cm longer each and a new square is made.

The area of new square is 19 cm² bigger than the original square.

Find the original square area.

(2) Three people of Taro, Jiro, Hanako took a test of Math.

The total score of three people was 228 points.

Taro's score was 18 points higher than Jiro's score.

The average score of Taro and Jiro was the same as a score of Hanako.

Find Taro's score.

(3) The price of pork 150g and beef 300g is 1500 yen.

The price of pork 250g and beef 150g is 1240 yen.

Find the price of pork 100g and beef 100g respectively.

Problem 3

There are P point and Q point 720m away.

Taro and Jiro left P at the same time and made a round trip between P and Q.

Jiro reached P three minutes later after Taro reached P.

The ratio of speed of Taro and Jiro is 3 : 2.

(1) Find the speed of Taro in meter per minute.

(2) Find the time of minutes and seconds when two people passed each other after they left P.

Problem 4

Some pieces of apple and pear were sold respectively at a fruit parlor.

On Day 1, 50% of the number of the whole apple, 40% of the number of the whole pear sold.

On Day 2, remaining apples and pears were sold.

All pears were sold.

As for the apple, only the same number was sold as pear sold on Day 1 and 42 remained.

As for the total number of apple and pear sold on Day 2, 18 decreased from the total number of two fruits sold on Day 1.

Find the each number of apple and pear in this parlor first.

Problem 5

There is a trapezoid ABCD whose side AD and BC are parallel as showing in the figure below.

E is a point on the side BC. AB and DE are parallel.

This trapezoid is divided into five triangle P, Q, R, S, and T with AC, DE, and BF.

The area of P is 32 cm².

The area of Q is larger than the area of S by 8 cm².

(1) Answer triangles whose area is equal among P, Q, R, S, and T.

(2) When the height of trapezoid ABCD is 16cm and an area is 280cm², find the length of the EC.

Problem 6

There are some number of two kinds of trapezoidal-shaped tile A and B shown as below respectively.

I will arrange A and B by connecting sides of A and B with 1cm in length and make figures such as Fig.1 and Fig.2.

For instance, as for Fig.1, it is a figure where three A tiles were arranged and Fig.2 is a figure where two A and two B were arranged.

Answer the following questions.

(1) I make a figure by arranging two A and six B.

Find the area ratio between the figure surrounded by inner sides of this figure and the figure surrounded by inner sides of Fig.2.

(2) I make a figure by arranging four A and two B.

Find the length of surroundings outside of the figure.

Problem 7

There is a bar of length 180cm.

I mark points on the bar where it is divided M equally and N equally respectively.

M and N are integers and M is bigger than 2 and is smaller than N.

For example when M is 4 and N is 6, it is marked respectively.

As the mark at one point is overlapped as shown in the lower figure, number of point becomes seven.

Answer the following questions.

(1) In case M is 5 and N is 6, find the number of point.

Find the shortest length of interval between two points.

(2) In case M is 8 and N is 12, find the number of point.

Find the shortest length of interval between two points.

(3) There are two cases of combination of M and N which make the number of point eight.

Answer these combinations of M and N.

(4) Answer all combinations of M and N that the shortest length becomes 12cm among the length of the interval between points.