Number riddle

RIDDLE

There are 5 positive whole numbers. The first is the square of the
second. The sum of the 2nd and the 3rd is 10. The 4th is greater than
the 2nd by 1. The sum of the 3rd and the 5th is 14. The sum of all the
numbers is 30. Find the values of the numbers.

SOLUTION

Represent the 5 numbers by any 5 letters of the alphabet you fancy. I
am using a, b, c, d and e for the 1st, 2nd, 3rd, 4th and 5th numbers
respectively.

From the statements in the question,

a = b² ................ (1)
b + c = 10 ................(2)
d = b + 1 ................(3)
c + e = 14 ................(4)
a + b + c + d + e = 30................(5).

* Express any 4 of the unknowns in terms of the 5th unknown in
equations (1) to (4).

a = b² (equation 1).
c = 10 - b (from equation 2).
d = b + 1 (equation 3).
e = 14 - c (from equation 4) = 14 - (10 - b) (from equation 2) = 14
- 10 + b = b + 4.

* Substitute these values into equation 5.

a + b + c + d + e = 30.

b² + b + (10 - b) + (b + 1) + (b + 4) = 30.

b² + 2b + 15 = 30.

b² + 2b + 15 - 30 = 0.

b² + 2b - 15 = 0.

(b + 5) (b - 3) = 0.

Either b + 5 = 0 or b - 3 = 0.

b + 5 = 0 ➡ b = - 5.

b - 3 = 0 ➡ b = 3.

Therefore, b = 3 (positive number).

Hence, a = b² = 3² = 9 ;
c = 10 - b = 10 - 3 = 7;
d = b + 1 = 3 + 1 = 4;
e = b + 4 = 3 + 4 = 7.

Therefore, the 1st, 2nd, 3rd, 4th and 5th numbers are 9, 3, 7, 4 and
7 respectively.

NOTE: * If you express the other unknowns in terms of a, substitution
of the values will lead you to an equation involving surds which you
can manipulate into a quadratic equation (by isolating the surd on one
side of the equation and then squaring both sides of the equation).
You will get 2 values of a (a = 9 or a = 25). Use the smaller value of
a , or your
a + b + c + d + e will be greater than 30 (violating equation 5).

How to draw a compound solid containing 2 cones with WPS Office, Polaris Office,etc on your smartphone.

Insert 2 triangles into the WPS Writer or Presentation document and drag them to meet along one of their sides. Screenshot the page to merge them together. Next, insert a circle and flatten it into a thin oval shape. Drag this thin oval shape to cover the intersection of the triangles. Use dotted lines to represent hidden-from-view lengths such as the base radius and the perpendicular heights of the compound solid. Take a screenshot of the page to merge the components into a single inseparable diagram. Use Photo Wonder or similar apps to insert text into the diagram.

Drawing compound shapes with WPS Office : Cone + cylinder.

Insert a triangle and a cylinder into the Office document. Send the triangle to the back of the cylinder after dragging the two shapes to meet along one of their edges. Stretch the triangle to the right size. Use dotted lines to represent hidden-from-view lengths such as the base radius and the perpendicular heights of the compound solid. Take a screenshot of the page to merge the components into a single inseparable diagram. Use Photo Wonder or similar apps to insert text into the diagram.

How to draw a cone with WPS Office on your smartphone

Insert a triangle into the WPS Writer or Presentation document. Next, insert a circle and flatten it into a thin oval shape. Drag this thin oval shape to cover the base of the triangle. Use dotted lines to represent hidden-from-view lengths such as the base radius and the perpendicular height of the cone. Take a screenshot of the page to merge the components into a single inseparable diagram. Use Photo Wonder or similar apps to insert text into the diagram.

Word problem

A box contains 13 currency notes, which are either ₦50 or ₦20 notes.
The total value of the currency notes is ₦530. How many ₦50 notes are
in the box?

SOLUTION

Let the number of ₦50 notes be x and the number of ₦20 notes be y.

Number of notes = x + y = 13................(1).
Total value of the notes = 50x + 20y = 530................(2).
Multiply (1) by 20.
20x + 20y = 260................(3).
Subtract equation (3) from (2).
30x = 270.
X = 9.
Therefore,
9 + y = 13 ................(1).
Y= 13 - 9 = 4.
Hence, there are 9 ₦50 notes (and 4 ₦20 notes) in the box.

................................................
CHECK: 9 + 4 = 13, and (50 x 9) + (20 x 4) = 450 + 80 = 530.

How to convert recurring decimals to common fractions

Converting recurring decimals to common fractions.

This requires a special process. Let the recurring decimal be equal to
a letter of the alphabet. Multiply both sides by 10 raised to power n,
where n is the number of recurring digits in the recurring decimal.
Subtract the first equation from the second and you will get the
fractional value of the recurring decimal.

EXAMPLES

(a) 0.111................

Solution.

R = 0.11... .............(1).
10R = 1.11... ................(2).
Subtract equation (1) from (2).
9R = 1.
R = 1/9.

(b) 0.1515................

Solution

X = 0.1515... .............(1).
100X = 15.1515... .............(2).
Subtract (1) from (2).
99X = 15.
X = 15/99 = 5/33.

(c)0.285714285714................

Solution

t = 0.285714... .............(1).
1 000 000 t = 285 714. 285714... .............(2).
Subtract (1) from (2).
999 999 t = 285 714.
t = 285 714 / 999 999 = 2/7.

How to draw Maths diagrams with WPS office.

Even if you don't have a computer ,you can still draw Maths diagrams
with WPS Office on your Android smartphone . Download Photo Wonder and
the latest version of WPS office (which has "Sharing by Screenshots"
on its Writer) from Google Play. Then follow the steps in the picture
below ( which was created using the same procedure I am about to teach
you).