Inequalities can be treated just like equations with one major difference - if you multiply or divide both sides using a negative number then the Inequality Sign is reversed. This doesn't actually apply in these questions. 1) 8x + 9 < 9x - 8...add 8 to both sides and subtract 8x from both sides 17 … Read more
As the expression is a radical (a root of another quantity) and figures within the square root are multiplied then see is any of them can be expressed as perfect squares. If this can be done then the square root of these perfect squares can be taken outside the square root sign. X2 is clearly … Read more
A 10 foot ladder is leaning against a house. The bottom of the ladder is 5 feet from the building. How high is the top of the ladder? How do I round to the nearest hundredth?
We can regard the ladder, the floor and the house wall as forming a right angled triangle with the ladder representing the hypotenuse. Using Pythagoras' Theorem and letting H be the height of the ladder then, 102 = 52 + H2 H2 = 100 - 25 = 75 H = √75 = 9.21954 (5dp) As … Read more
The sum of the first n positive integers is equal to 1/2(n squared+n). How many of the first n positive integers must be added to give a sum of 325?
The required equation is, 325 = 1/2(n2 + n)...then 650 = n2 + n...and the equation can be written:- n2 + n - 650 = 0...which can be factored (n + 26)(n - 25) = 0 As we are only concerned with a positive integer solution, this occurs when n - 25 = 0, thus … Read more
A retail jeweler sells a pair of gold earrings for $213.90. The jeweler uses a 130% markup based on cost. Based on this information, what was the wholesale cost of the earrings?
The wholesale cost can be regarded as 100%. After a markup of 130% the retail price is equivalent to 230% (100 + 130) of the wholesale cost. If the retail price = 213.90 then the wholesale price = 100 x (213.90/230) = $93.00
How do I solve by rearranging the equations first. Then use the multiplication principle and then use the elimination method: 3x=8y+11 x+6y-8=0?
3x = 8y + 11 x + 6y - 8 = 0...which can be written as x = -6y + 8 We now have both equations in the same format. It can be seen that we can make the coefficients of x equal in both equations by multiplying the 2nd equation by 3. 3x = … Read more
The formula to calculate the present amount including compound interest is A = P(1 + r/n)nt , where P is the principal amount, r is the annual rate expressed as a decimal , t is the number of years, and n is number of times per year that interest is compounded. NOTE : It is … Read more
If , a/b = 7/20 then 20a = 7b and we can write... 20a - 7b = 0 As a + b = 135 then multiplying throughout by 7 gives 7a + 7b = 945 Adding these two equations gives, ...27a = 945 : Then a = 945/27 = 35 Substituting this value for a … Read more
The Slope/Intercept form of equation for a straight line is y = mx + c, where m is the slope(gradient) and c is the y-intercept (when x = 0). 4x - 10y = 11 10y = 4x - 11 y = 4x/10 - 11/10 y = 2x/5 - 11/10 (or y = 0.4x - 1.1)
The dimensions of a rectangle are such that its length is 5 in. More than its width. If the length were doubled and if the width were decreased by 2 in., the area would be increased by 162 in. What are the length and width of the rectangle?
Let the width of the original rectangle be W, then the length = (W + 5) The area of this rectangle = W(W + 5) = W2 +5W The modified rectangle has a length 2(W + 5) and a width (W - 2) The area of the modified rectangle = 2(W + 5)(W - 2) … Read more
The polynomial of degree 4, P(x) has a root of multiplicity 2 at x=3 and roots of multiplicity 1 at x=0 and x=-4. Its lead coefficient is 3. Find a formula for P(x)?
The factored formula for the polynomial is, (x - 3)(x - 3)x(x + 4) = 0, given that it has two equal roots when x = 3, a further root when x = 0 and another when x = -4. This expands to x4 - 2x3 - 15x2 + 36x = 0 As the lead … Read more
Joe is trying to soup up his dragster He knows that the time needed for the car to go from 0 to 100 miles per hour varies inversely with the cars horsepower At 200 horsepower the car can go from 0 to 100 mph in 10 seconds How long should the car?
The question appears incomplete but certain facts can be deduced. As Time (T) varies inversely with Horsepower (P) then, T ∝ 1/P which implies T = k/P where k is a constant. When T = 10, P = 200 therefore , 10 = k/200 : K = 200 x 10 = 2000 The formula now … Read more
(w-8/w-9) - (w+1/w-9) + (w-27/w2-81) As the first two terms have the same denominator they can be combined : (w-8/w-9) - (w+1/w-9) = -9 / w - 9 The denominator for the third term can be factored :- (w-27/w2-81) = w - 27 / (w + 9)(w - 9) Combining the new expressions gives :- … Read more
John is 13 years less than 3 times as old as his son. Six years ago, he was 14 years more than twice as old as his son was then. How old is John?
Let John's age now be J and his son's current age be S. The looking at the ages now, J + 13 = 3S : Thus S = (J + 13) ÷ 3 And looking at 6 years ago, (J - 6) - 14 = 2(S - 6) : J - 20 = 2S - … Read more
The equation of the circle can be written (x - p)² + (y - q)² = r², where r is the radius and (p,q) is the centre of the circle. Then, (x - 3)² + (y + 1)² = 9², is the equation of the circle with its centre at (3, -1)and radius 9.
A boy's mother is a chemist and his father is a mathematician. His nickname is "iron59". Now what is his real name?
FELIX Fe is the Chemical Symbol for Iron LIX is 59 in Roman Numerals.
What 5-digit number has the following features: If we put the numeral 1 at the beginning, we get a number three times smaller than if we put the numeral 1 at the end of the number?
Let the 5 digit number be represented by abcde. Then the 6 digit number, 1abcde x 3 = abcde1 (another 6 digit number) Now let Y = abcde.00 Then (100000 + Y) x 3 = (Y x 10) + 1 300000 + 3Y = 10Y + 1 7Y = 299999 Y = 299999 ÷ 7 … Read more