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Sorry! I missed the last clue. The largest such number would be:94320

The largest such number =85,320.

I found all the y , just need help finding the x's

The smallest positive 5-digit integer =10236

I'm getting 12384, which is divisible by 1, 2, 3, 8, and 4.

Thank you so much!!!

Using the quadratic equation to find the solutions to  7x² -x - 3  =  0, we can get these answers:

a  =  [ 1 + sqrt(85) ] / 14       and     b  =  [ 1 - sqrt(85) ] / 14 

Both  1 + a + a² + a³ + ...     and    1 + b + b² + b³​ + ...     

are  infinite geometric series with common ratios in the range  -1 < r < 1,

so we can use the formula:  Sum  =  a₁ / ( 1 - r)

In both cases,  a₁ = 1, so:

the sum of  1 + a + a² + a³ + ...  is  1 / [ 1 - (1 + sqrt(85))/14 ]  

     =  14 / [ 14 - (1 + sqrt(85) ]     =     14 / [ 13 - sqrt(85) ]

and the sum of 1 + b + b² + b³​ + ...   is  14 / [ 13 + sqrt(85) ]

Multiplying these two values together, we get  196 / (169 - 85)   196 / 84  =  7/3

2+x/x = 5/3

cross multiply

3(2+x)=5x

distribute

6+3x=5x

6+3x-3x=5x-3x

6=2x

6/2=2x/2

x=3

The runners' speed was 3 miles per hour and the cyclists' speed was 5 miles per hour

Let the number of participants =P
2/3 * P= 54, solve for P

P = 81 - Total number of people[students and chaperones]
81 - 7 =74 - seventh grade students went on the field trip.

Thanks Heureka :)

I absolutely agree.

You should always test or think about whether an answer is correct or reasonable.

Sometimes in exams, there may not be much time for checking but even under exam conditions, a quick check should be done.

Let the of single persons =P
Let the number of couples =C

3P + 5C =475,...................(1)
P + 2C   =180....................(2), solve for P, C

P =50 - single persons
C =65 - couples x 2 =130 persons

An engineer invested a portion of $15,000 in a 7% annual simple interest account and the remainder in a 6.5% annual simple interest government bond. The amount of interest earned for one year was $1020. How much was invested in each account?

Let the amount invested @ 7% =M

The amount invested @ 6.5%   =$15,000 - M

0.07M + 0.065[15000 - M] =1020, solve for M

M = $9,000 invested @ 7%

$15,000 - $9,000 =$6,000 invested @ 6.5%.

We can solve for y in the equation x + 5y = 7 to get y = -1/5*x + 7/5, so the slope of the line is -1/5.

Then the perpendicular line will have slope -5.  We want the perpendicular line to go through (-8,3), so by point-slope form, the equation of the line is y - 3 = -5(x + 8).  Expanding, we get y = -5x - 37, so 5x + y = -37.

We want the coefficient of x to be 15, so we multiply by 3: 15x + 3y = -111.  Therefore, (A,B) = (3,-111).

There are: 810 "Swappy Numbers" if you count the 90 palindromes that are already there. If you don't wish to include them, then you have: 810 - 90 =720 "Swappy Numbers"

Now that I think about it, why do p+q+r=7?

0 + 1 + 7 + 6 + 1  = 15 ways - with no restrictions. This is Bell's 4th number, which is the sum of Stirling Numbers of the 2nd Kind(4, 0), (4,1), (4,2), (4,3) and (4,4). 

In the diagram below, triangle ABC is isosceles, and triangle MPQ is equilateral. Find the area, in cm2, of triangle MPQ.

BM = 36 cm

AC = BC

PC = QC

Let an F to be the foot of altitude from M to BC

∠FMP = 15°

FM = sqrt [( 36² )/2] = 25.455844

MP = FM / cos(15°) = 26.353829

PQ/2 = MP/2 = 13.1769145

The height of ΔMPQ is H = tan(60°) * (PQ/2) = 22.8230854

Area of  Δ MPQ = (PQ/2) * H = 300.7378467 cm²  

Future population =FP

Present population =PP

FP = PP x [1 + R]^N

88,000 =22,000 x [1.12]^N, solve for N

4 = 1.12^N

N =Log(4) / Log(1.12)

N =12.23 Years for the population to quadruple.

most appreciated!!

3950 / 5 =790 km/h - its speed against the wind
3030 / 3 =1,010 km/h - its speed with the wind.

[1,010 + 790] / 2 =900 km/h - its speed in still air.
[1,010 - 790] / 2 =110 km/h - The speed of the wind.

1 = F + D*(4 - 1),   
-2=F + D*(10 - 1), solve for F, D

F =5/2 - First term.
D = - 1/2 - Common difference.

a(n) =[6 - n] / 2 ...... and your sequence contiues....

5/2, 2, 3/2, 1, 1/2, 0, -1/2, -1, -3/2, -2, -5/2, -3, -7/2, -4, -9/2, -5, -11/2, -6, -13/2, -7, ...
     

Your N = 315 divisors, 18 of which are Perfect Cubes:

So, the probability is: 18 / 315 = 2 / 35

The difference of two numbers is 6 and the sum of their squares is 68. Find the numbers. 

X –  Y  =  6 

X² + Y²  =  68 

Without doing lengthy calculations, I recognized that the larger number has to be 8 or less.

So I said, 64 and what equals 68 and the answer was 4.  The sqrt(4) is 2. 

8 – 2  =  6  so I happened to get it right on the first try.  The numbers are 8 and 2. 

If 8 hadn't worked I'd have tried 7.  49 and what equals 68, the answer is 19. 

19 isn't a perfect square so this pair fails.  Moving on to 6.  36 and 32 don't work. 

nor do 25 and 43,  nor do 16 and 52.  nor 9 and 59. 

the next one to try is 4 and we already know that 2 is one of the numbers.  We're done. 

_.

Let   T  be the cost of a single taco

We have this

4T + 2(2.00)  = 2(T)  + 4(2.00)     simplify

4T  +  4  = 2T  +   8            subtract  4, 2T  from both sides

2T    = 4                   divide both sides  by   2

T  =  2     →     one taco costs   $2