Tutor MathsGuru: Ask me for your burning Maths questions!

[Vanilla Cake]

Thks to Maths Hub and iFruit for your detailed and clear solutions. Sorry, some more questions from APMOPS 2005 that need your help.


Q2
How many whole numbers from 1 to 1000 can be expressed as the difference of the squares of two whole numbers?
[Note: 0 is a whole number.]

Q3
The following number is made up of all the digits of the whole numbers 1 to 2005.
12345678910111213141516..........20042005
Find the number of zeros in this number.

Q6
Allen , Benedict and Carl started at the same instant from the same point using the same route trying to overtake a fourth cyclist Donald traveling at a constant speed ahead of them. Allen and Benedict each took 10 hours and 2 hours respectively to overtake Donald.
Given that Allen , Benedict and Carl each cycled at the constant speed of 4 km/h, 5 km/h and 10 km/h respectively throughout the journey, find the time, in hours, that Carl took to overtake Donald.

Source: http://www.hci.sg/aphelion/apmops/2007/pdf/English/2005%20English%20IR.pdf.

Could you pls advise the correct answers for Q1, Q4 and Q5 too?
Thks and much appreciated for your help.

[iFruit]

Vanilla Cake:

Thks to Maths Hub and iFruit for your detailed and clear solutions. Sorry, some more questions from APMOPS 2005 that need your help.


Q2
How many whole numbers from 1 to 1000 can be expressed as the difference of the squares of two whole numbers?
[Note: 0 is a whole number.]

we know a² - b² can be written as (a+b)(a-b) i.e. product of the sum of the two numbers and difference of the two numbers.

1) Assume a, b are both even or both odd. Then a+b is even and a-b is even. so (a+b)(a-b) is a multiple of 4.

That means every multiple of 4 can be written as a difference of two squares.

2)Assume one of a,b is odd and the other even. Then a+b is odd and a-b is odd
so (a+b)(a-b) is a odd number. For example it can be written as

(a+b)(a-b) = 2n+1 where (a+b)=2n+1, (a-b) = 1 giving a = n+1, b =n

That is every odd number can be written as a diff of two squares.


So all even numbers that can be divided by 4 and all odd numbers can be expressed as a sum of squares. Only even numbers that can be divided by 2 but not by 4 can't be expressed in a such a manner.

so there are 500 even numbers between 1..1000 of which only 250 can be expressed divided by 2 but not by 4.

So 750 numbers can be expressed as difference of squares.

[iFruit]

Vanilla Cake:

Thks to Maths Hub and iFruit for your detailed and clear solutions. Sorry, some more questions from APMOPS 2005 that need your help.


Q3
The following number is made up of all the digits of the whole numbers 1 to 2005.
12345678910111213141516..........20042005
Find the number of zeros in this number.

Corrected following maths hub solution.. Thank Maths hub!

1) we know between 2000 .... 2005 there are 13 zeros

2) Now consider numbers 1000 ... 1999

The 1000's digit is always one so 0 can appear only in 100's, 10's or 1's places.

If there is only one 0, in those numbers, the other two places can vary between 1 to 9 giving 9x9 = 81 possibilities. Since there are three places we could have 3x81 single zeros between 1000...1999

Similarly for two zeros, we will have 3x9 times number 0s appearing = 27x2 0's

3 0's (number 1000 ) = 1 time = 3 0's

3) Consider numbers 100 to 999

we get 2x 81 numbers with one zero, 9 numbers with two zeros (18 zeros)

4) Consider numbers 10..99

we can have 0 in 1's digit place 9 times

So that total number of zeros = 13 + 3x81 + 27x2 + 3 + 9x2 + 2x81 +9 = 502

[iFruit]

Vanilla Cake:

T

Q6
Allen , Benedict and Carl started at the same instant from the same point using the same route trying to overtake a fourth cyclist Donald traveling at a constant speed ahead of them. Allen and Benedict each took 10 hours and 2 hours respectively to overtake Donald.
Given that Allen , Benedict and Carl each cycled at the constant speed of 4 km/h, 5 km/h and 10 km/h respectively throughout the journey, find the time, in hours, that Carl took to overtake Donald.

Allen = A, Benedict = B, Carl = C, Donald = D.

Assume the initial distance between A,B,C and D is d
speed of D = s km/h

So time taken for A to meet D = d/(4-s) = 10 hours ....(1)
time taken for B to meet D = d/(5-s) = 2hours.....(2)
time taken for C to meet D = d/(10-s) = ?

for (1) and (2), we get 10(4-s) = 2(5-s) ---> 20 - 5s = 5 -s ---> s = 15/4

Then d = 10( 4 -s ) = 10 (4 - 15/4) = 10/4 km

time taken for C to meet D = d/(10-s) = (10/4) / (10 - 15/4) = 10/25 = 2/5 hours.

HTH

[iFruit]

deleted

[Maths Hub]

Vanilla Cake:

Thks to Maths Hub and iFruit for your detailed and clear solutions. Sorry, some more questions from APMOPS 2005 that need your help.


Q3
The following number is made up of all the digits of the whole numbers 1 to 2005.
12345678910111213141516..........20042005
Find the number of zeros in this number.

From 1 to 99-->'0' appears 9 times.
From 100 to 199-->'0' appears 20 times.
Similarly, from 200 to 299 -->'0' appears 20 times.

So from 100 to 999 --> '0' appears 20x9=180 times

From 1000 to 1999 --> '0' appears 100 times in the hundred digit place, 100 times in the ten digit place, 100 times in the ones digit place. Hence total = 300

From 2000 to 2005 --> '0' appears 13 times.

Total '0': 9+180+300+13=502

[noob]

Need solution for following question, please help tks


Alice, Betty and Cecilia shared the cost of a present. Alice paid 1/3 of the total amount of money paid by Betty and Cecilia. Betty paid 1/8 of the amount of money paid by Alice and Cecilia. Alice paid $60 less than Cecilia.
(a) How much did Betty pay?
(b) What was the cost of the present?

[Daddy]

Hi,


Need help!

1. Chloe has a box of beads. 2/7 of the beads are white and the rest are blue and orange beads. The ratio of the number of white beads to the number of blue beads is 6:5. find the ratio of the number of blue beads to orange beads. Give your answer in the simplest form.

Thanks…
Daddy.

[iFruit]

Daddy:

Hi,


Need help!

1. Chloe has a box of beads. 2/7 of the beads are white and the rest are blue and orange beads. The ratio of the number of white beads to the number of blue beads is 6:5. find the ratio of the number of blue beads to orange beads. Give your answer in the simplest form.

Thanks..
Daddy.

2/7 beads are white. So White: (blue+orange) = 2:5 = 6:15

white: Blue = 6:5

Observing the above two ratios we deduce

whilte:blue:orange = 6:5:10

So blue:orange = 5:10 = 1:2

[iFruit]

noob:

Need solution for following question, please help tks


Alice, Betty and Cecilia shared the cost of a present. Alice paid 1/3 of the total amount of money paid by Betty and Cecilia. Betty paid 1/8 of the amount of money paid by Alice and Cecilia. Alice paid $60 less than Cecilia.
(a) How much did Betty pay?
(b) What was the cost of the present?

Is there a typo in the question? Perhaps \"Alice paid $70 less than Cecilia\"?

Below is working assuming \"Alice paid $70 less than Cecilia\". With $60, it is the same working but we end up with all fractions

Alice = A, Betty = B and Cecilia = C.

A = 1/3 (B +C) ----> 3A = B+C ---------(1)
B = 1/8 (A+C) -----> 8B = A+C ---------(2)
C = A + 70---(3)

Substituting (3) in (1) and (2) we get

3A = B+A+70 -----> 2A = B+70
8B = A +A +70----> 8B = 2A +70

From the above two we get, 8B = B+70+70 or B = 20

The A = 45, C = 115, Total = 115+45+20 = 180

HTH