maths

All the students in a school are made to stand in rows so as to form an equilateral triangle. The first row consists of one student, the second row 2 students, third row 3 students and so on. Had there been 90 more students, the total number of students could have been arranged in the shape of a square, so that each side of the square has 5 students less than the number of students in the side of the triangle. Find the total number of students initially present in the school?

  1. 👍
  2. 👎
  3. 👁
  1. You are dealing with the triangular numbers whose sums of consecutive rows would be

    1 3 6 10 15 21 28 ...

    these numbers are generated by (n^2+n)/2 for n equal to the set of natural numbers.

    So I added 90 to each of these to see if I get a perfect square.
    The first such n value was at n=4
    the sum of students would have been 10, and adding 90 gave me 100.
    But that did not satisfy the second condition you gave, namely that the side of the square had to be 5 less than the side of the triangle.

    I kept going for a while, almost gave up, but came to n = 26

    the sum of the students in the triangle is (26^2 + 26)/2 = 351
    adding 90 got me 441 which is 21^2
    and the difference between these is indeed 5

    so there were initially 351 students in the school

    1. 👍
    2. 👎
  2. as first row consist 1 student, 2nd row consist 2 students and so on...
    let the no of rows in triangle be "n" now the total no of students in triangle will be 1+2+3+....+n=n(n+1)/2
    given if 90 more students are added that is n(n+1)/2+90 is equal to no of students in square arranged in such a manner that has 5 less students each side than the triangle.
    so now n(n+1)/2+90=(n-5)^2 (area of the square )
    n^2-21n-130=0
    n=26 substituting in n(n+1)/2 we get 351.
    answer is 351

    1. 👍
    2. 👎

Respond to this Question

First Name

Your Response

Similar Questions

  1. Math

    The first row of a school concert hall has 25 seats and each row after the first has one more seat than the row before it there are 32 rows of seats write the rule or equation for the number of seats in the nth row 35 students

  2. Math

    The school band is comprised of middle school students and high school students, but it always has the same maximum capacity. Last year the ratio of the number of middle school students to the number of high school students was

  3. maths

    there are 104 students in class 10th and 96 students in class 9th in a school .in a house examination the students ate to be evenly seated in parallel rows such that no two adjacent rows are of the same class. (a) find the maximum

  4. Math

    You want to find out what the favorite hot lunch in the school cafeteria is among the high school students. At an assembly for the whole school, you decide to survey all students who are sitting on the end of their rows in the

  1. math

    The Green County school system has 2,997 high school students, 3,831 middle school students, and 5,084 elementary school students. How many total students are in the Green County school system? is it 11912 ?

  2. Math

    Barbara has a mixed garden that has 16 rows of different flowers and vegetables. One fourth of the rows are lettuce, 1/8 of the rows are pumpkins, and 1/2 of the rows are red tulips. The other rows are carrots. How many rows of

  3. MATH

    A researcher wants to select a sample of 50 students from four local private high schools by performing stratified sampling. The enrollments are shown in the table. How many students at each school should be included in the​

  4. Math pls help me

    A dance school has 54 students who learn salsa and 23 of those students also learn ballet. There are 15 students who do not learn salsa but learn ballet and 10 students do not learn either salsa or ballet. Which table best shows

  1. Math

    1. Trevor has $65 to spend at the sporting-goods store. He buys 3 cans of tennis balls for $6 each. How many $8 sports towels can Trevor buy? a. 3 towels b. 5 towels ******* c. 6 towels d. 8 towels 2. There are 39 students in the

  2. math

    There are 154 seats in the school auditorium. There are 7 seats in each row. There are 70 people seated, filling up full rows of seats. How many rows are empty?

  3. math

    At Murky Middle School, 372 students ride the bus to school. If this number is 60% of school enrollment, then how many students are enrolled at the school?

  4. math

    A sixth-grade class has 20 total students, while the entire middle school has 1,000 students. A census is taken of the entire school’s population to see on which day of the week every student was born. Theoretically, 14.3%, or

You can view more similar questions or ask a new question.