NMTC Stage 2 GAUSS Primary Level 2017

If $$\frac{1}{5\frac{1}{3}} + \frac{1}{3\frac{3}{7}} + \frac{1}{4\frac{4}{7}} + \frac{1}{?} = \frac{77}{96}$$, find what should be filled in the place marked.

There are 10 cards numbered 1 to 10. There are three second standard children Ram, Bilal and Cynthia. The teacher selects 3 cards from the 10 cards without seeing the numbers. She distributes the cards to the children one to each. After the children note down the numbers on the cards she collects them back. She repeats the same process two more times, so each child now has 3 numbers noted down. The teacher asks them to add the numbers and tell her the sums obtained. They told her the sums were 10, 14, 15, but Ram received the same card three times. Bilal and Cynthia received all cards different. What numbered cards did each receive?

In the adjoining figure ABCD is a rectangle. Points P, Q, R, S are marked as in the diagram such that $$AP=PQ=QB$$. R is the midpoint of CD. If $$AS \colon SD = 3 \colon 1$$, find the ratio of the areas of triangle ASP, quadrilateral SPRD, triangle PQR and the trapezium QBCR.

Take the numbers 1, 2, 3, 4, 5, 6, 7 and 8. We have to make two groups, A and B, each containing four numbers such that 

(a) The sum of the numbers in group A is equal to the sum of the numbers in group B, 

(b) Group A has a number such that when it is moved from group A to group B, the sum of the five numbers in group B is equal to twice the sum of the 3 numbers in group A, and 

(c) Group B has a number such that when it is moved to group A, the sum of the three numbers in group B is $$\frac{5}{7}$$ of the sum of the 5 numbers in group A. Find the numbers in groups A and B.

Mahadevan was puzzled by the strange way in which his granddaughter was counting. She began to count on the fingers of her left hand, calling the thumb 1, the first finger 2, middle finger 3, ring finger 4, little finger 5, then reversed direction, calling the ring finger 6, middle finger 7, first finger 8, thumb 9, then back to the first finger for 10, middle finger for 11, and so on. She continued counting back and forth until she reached a count of 20 on her ring finger. Seeing this, Mahadevan told her that if she could find on which finger she would count 2017, he would buy her an ice cream. Can you find on which finger she will count 2017? Explain the steps.

In the adjoining figure the number in each circle is the sum of the numbers in the two adjacent circles below it. 

(a) Find X, writing the steps systematically. 

(b) What is the least positive number to be added to X so that the result is a perfect square? 

(c) What is the least positive number to be subtracted from X so that the result is a perfect square?