AMO 2027 Grades 4-5 SET 13 - Self-study

Evaluate:

(a) $\frac1{5\times8}+\frac1{8\times11}+\frac1{11\times14}+\cdots+\frac1{98\times101}$.

(b) $\frac12-\frac5{12}+\frac7{24}-\frac9{40}+\frac{11}{60}-\frac{13}{84}+\frac{15}{112}-\frac{17}{144}$.

Answer entry: (a);(b), using simplified fractions a/b, separated by semicolons.

**Editorial input note:** Use no spaces before or after semicolons.

A book has 500 pages numbered 1, 2, 3 and so on. How many times does the digit 1 appear in the page numbers?

The bar graph below shows the number of marbles each child has. Study it carefully and answer the questions that follow:

(a) Who has twice as many marbles as Alan?

(b) How many fewer marbles does Nigel have than Samad?

(c) How many marbles must Samad give Nigel so that they will have the same number of marbles?

Readable graph data: Alan 110, Samad 220, Kelly 140, Nigel 80 marbles. The vertical axis is Number of marbles, with tick spacing 20.



Answer entry: (a) name;(b) number;(c) number, separated by semicolons.

**Editorial input note:** Use no spaces before or after semicolons.

In a division statement, the quotient is 5 and remainder is 6. The difference between the divisor and the dividend is 38. Find the values of the divisor?

In the diagram below, the diagonals of quadrilateral ABCD intersect at E. Given that the areas of triangles ADE, ABE, BCE and CDE are 75 cm², 45 cm², 60 cm² and x cm², find x.

Find the value of $\diamond$.

$\diamond+\square+\circ+\diamond=33$

$\square+\circ+\diamond+\circ=26$

$\square+\circ+\square+\diamond=41$

The sum of five consecutive numbers is 30. What is the largest number in these numbers?

Four teams are competing against one another for District Volleyball Championship. Every team must play exactly once against the other three teams. So far, Team A has played 3 matches, Team B has played 2 matches and Team D has played 1 match. How many matches has Team C played?

What is the sum of all the digits of the page numbers of a 90-page book?

Isabella had a week to read a book for a school assignment. She read an average of 36 pages per day for the first three days and an average of 44 pages per day for the next three days. She then finished the book by reading 10 pages on the last day. How many pages were in the book?

(a) A boy runs 200 metres in 24 seconds. What is his speed in km/h?

(b) If the speed of a man is 45km per hour, then what is the distance traveled by him in 30 seconds?

(c) A train 575m long crosses a tunnel of length 325m in 90 sec. What is the speed of the train in km/h?

(d) A train traveling at 60km/h crosses a man in 6 seconds. What is the length of the train in metre?

(e) A car takes 4 hours to cover a distance, if it travels at a speed of 40 m/h. What should be its speed to cover the same distance in 16 hours?

(f) How long will a boy take to run round a square field of side 35 meters, if he runs at the rate of 9 km/h? (in second)

(g) A train 150 m long is running with a speed of 54 km per hour. In what time will it pass a telegraph post?

(h) A train passes through a telegraph post in 9 seconds moving with a speed of 54 km per hour. What is the length of the train in metre?

Conventions: the man in (d) is stationary; trains fully clear the person/post/tunnel. In (e), retain the printed m/h (metres per hour), not km/h.

Answer entry: (a)–(h), in order, with units km/h; m; km/h; m; m/h; seconds; seconds; m. Enter only the eight numbers, separated by semicolons.

**Editorial input note:** Use no spaces before or after semicolons.

There are three ropes. The first ropes is 7m longer than the second rope. The second rope is 8m longer than the third one. If the total length of these three ropes is 95m, how long is each rope?

Answer entry: first;second;third rope, all in metres, separated by semicolons.

**Editorial input note:** Use no spaces before or after semicolons.

When I open my mathematics book, there are two pages which face me and the product of the two page numbers is 1806. What are the two page numbers?

Answer entry: smaller page;larger page, separated by semicolons.

**Editorial input note:** Use no spaces before or after semicolons.

A thick dowel stick has to be cut into eighteen pieces. Each cut takes you 30 seconds. How long will it take to cut the dowel stick into pieces?

Answer entry: seconds. Use separate cuts through one piece at a time, without stacking.

Jane has 360 ribbons. 15% of them are red ribbons and the rest are blue ribbons. She buys some more red ribbons and the percentage of red ribbons is increased to 40%. How many ribbons does Jane have after buying the red ribbons?