Method I

Let x = the total land

Of means multiplication

Tony shared $\frac{2}{3}$ of his plot of land

Land shared = $\frac{2}{3}$ x x = $\frac{\mathrm{2x}}{3}$

Plot of land received by each = $\frac{\mathrm{2x}}{3}$ ÷ 3

Every number is divisible by 1.

3 = $\frac{3}{1}$

Plot of land received by each = $\frac{\mathrm{2x}}{3}$ ÷ $\frac{3}{1}$

Reciprocate (the numerator becomes denominator and the donominator becomes the numerator) the fraction after the division sign and change the division sign to multiplication.

Plot of land received by each = $\frac{\mathrm{2x}}{3}$ x $\frac{1}{3}$

Plot of land received by each = $\frac{\mathrm{2x\; x\; 1}}{\mathrm{3\; x\; 3}}$

Plot of land received by each = $\frac{\mathrm{2x}}{9}$

Fraction = $\frac{\mathrm{Number}}{\mathrm{Total}}$

Fraction of plot of land received by each = $\frac{\mathrm{2x}}{9}$ ÷ x = $\frac{2}{9}$

Method II

Fraction for the entire plot of land = 1

Of means multiplication

Tony shared $\frac{2}{3}$ of his plot of land

Fraction of land shared = $\frac{2}{3}$ x 1 = $\frac{2}{3}$

Fraction received by each = $\frac{2}{3}$ ÷ 3

Every number is divisible by 1.

3 = $\frac{3}{1}$

Fraction received by each = $\frac{2}{3}$ ÷ $\frac{3}{1}$

Reciprocate (the numerator becomes denominator and the donominator becomes the numerator) the fraction after the division sign and change the division sign to multiplication.

Fraction received by each = $\frac{2}{3}$ x $\frac{1}{3}$

Fraction received by each = $\frac{\mathrm{2\; x\; 1}}{\mathrm{3\; x\; 3}}$

Fraction received by each = $\frac{2}{9}$

$\frac{9}{5}$,1.88,$\frac{15}{8}$

Method I

Change the fractions to decimal and compare them.

9 divided by 5

15 divided by 8

$\frac{9}{5}$ = 1.8

$\frac{15}{8}$ = 1.875

Ascending means from the smallest to the largest.

Arranging 1.8,1.88,1.875 ascending order = 1.8,1.875,1.88

Replace the decimals by their corresponding fractions in 1.8,1.875,1.88.

Ascending order = $\frac{9}{5}$,$\frac{15}{8}$,1.88

Method II

Change the decimal to fraction and compare the fractions

Every number is being divided by 1.

1.88 = $\frac{1.88}{1}$

Count how many digits (n) are after the decimal point and multiply both the numerator and the denominator by 1 joined by the n number of zeros.

In 1.88, there are two digits (8 and 8) after the decimal point and so we will multiply both the numerator and the denominator by 100. Multiplying the numerator by 100 changes the decimal to a whole number.

$\frac{1.88}{1}$ = $\frac{\mathrm{1.88\; x\; 100}}{\mathrm{1\; x\; 100}}$ = $\frac{188}{100}$ = $\frac{47}{25}$

4 divides 188, 47 times and 100, 25 times.

Now compare the fractions.

$\frac{9}{5}$,1.88,$\frac{15}{8}$

1.88 = $\frac{47}{25}$

$\frac{9}{5}$,$\frac{47}{25}$,$\frac{15}{8}$

To compare the fractions, the denominators must be the same. Hence find the equivalent fractions for each of the fractions having the same denominator. To do so follow the steps below.

1. Find the Lowest Common Factor (L.C.M) of the denominators. The L.C.M of 5,25 and 8 is 200.

2. Multiply the numerator and denominator of each fractions by how many times the denominator will divide the L.C.M

For the fraction with denominator 5, we have to multiply both the numerator and the denominator by 40 since 5 x 40 = 200
For the fraction with denominator 25, we have to multiply both the numerator and the denominator by 8 since 25 x 8 = 200
For the fraction with denominator 8, we have to multiply both the numerator and the denominator by 25 since 8 x 25 = 200

Equivalent Fractions

$\frac{9}{5}$ = $\frac{\mathrm{9\; x\; 40}}{\mathrm{5\; x\; 40}}$ = $\frac{360}{200}$

$\frac{47}{25}$ = $\frac{\mathrm{47\; x\; 8}}{\mathrm{25\; x\; 8}}$ = $\frac{376}{200}$

$\frac{15}{8}$ = $\frac{\mathrm{15\; x\; 25}}{\mathrm{8\; x\; 25}}$ = $\frac{375}{8}$

Now since the equivalent fractions have the same denominator, you can compare their numerators.

Ascending order of the numerators = 360, 375, 376.

Now replace the order of the numerators by the fractions or decimal that those numerators were obtained from.

360 was obtained from $\frac{9}{5}$

375 was obtained from $\frac{15}{8}$

376 was obtained from 1.88

Ascending order = $\frac{9}{5}$,$\frac{15}{8}$,1.88

$($ 1$\frac{1}{2}$ - $\frac{5}{6}$ $)$ x $\frac{9}{10}$

Change the mixed fraction to improper fraction.

1$\frac{1}{2}$ = $\frac{\mathrm{2\; x\; 1\; +\; 1}}{2}$ = $\frac{\mathrm{2\; +\; 1}}{2}$ = $\frac{3}{2}$

$($ 1$\frac{1}{2}$ - $\frac{5}{6}$ $)$ x $\frac{9}{10}$ = $($ $\frac{3}{2}$ - $\frac{5}{6}$ $)$ x $\frac{9}{10}$

$($ 1$\frac{1}{2}$ - $\frac{5}{6}$ $)$ x $\frac{9}{10}$ = $($ $\frac{\mathrm{3\; x\; 3\; -\; 1\; x\; 5}}{6}$ $)$ x $\frac{9}{10}$

$($ 1$\frac{1}{2}$ - $\frac{5}{6}$ $)$ x $\frac{9}{10}$ = $($ $\frac{\mathrm{9\; -\; 5}}{6}$ $)$ x $\frac{9}{10}$

$($ 1$\frac{1}{2}$ - $\frac{5}{6}$ $)$ x $\frac{9}{10}$ = $($ $\frac{4}{6}$ $)$ x $\frac{9}{10}$

$($ 1$\frac{1}{2}$ - $\frac{5}{6}$ $)$ x $\frac{9}{10}$ = $\frac{4}{6}$ x $\frac{9}{10}$

Note: 2 divides 4, 2 times and 6, 3 times

$($ 1$\frac{1}{2}$ - $\frac{5}{6}$ $)$ x $\frac{9}{10}$ = $\frac{2}{3}$ x $\frac{9}{10}$

Note: 2 divides itself once and 10, 5 times. 3 divides itself once and 9, 3 times

$($ 1$\frac{1}{2}$ - $\frac{5}{6}$ $)$ x $\frac{9}{10}$ = $\frac{1}{1}$ x $\frac{3}{5}$

$($ 1$\frac{1}{2}$ - $\frac{5}{6}$ $)$ x $\frac{9}{10}$ = $\frac{3}{5}$

Change the mixed fractions to improper fractions

Rule

Apply the division of fraction rule

Thus you reciprocate (down goes top and top down) the fraction at the right and change the division sign to multiplication

Now cancel out and apply the principle of multiplying fractions

3 goes into 6, 2 times and 21, 7 times

5 goes into itself, 1 time and 10, 2 times

Make sure each of the fractions have the same denominator

Steps

1. Look for the L.C.M for the denominators (5,4 and 3)

2. Multiply the numerator and the denominator by the number that when you multiply the denominatory with you will get the L.C.M

3. Once the denominators are the same, arrange the fraction using the values of the numerators

STEP 1

The L.C.M of 5, 4 and 3 is 60

The L.C.M is the least number you can find in the multiplication of each of the numbers you are looking for the L.C.M

Multiplication of 5

5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,...

Multiplication of 4

4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,...

Multiplication of 3

3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,...

As you can see the least number you can find in the multiplication table of 5,4 and 3 is 60

STEP 2

Denominator 5

5 x 12, gives 60, hence we have to multiply the numerator (3) and the denominator (5) by 12

Denominator 4

4 x 15, gives 60, hence we have to multiply the numerator (1) and the denominator (4) by 15

Denominator 3

3 x 20, gives 60, hence we have to multiply the numerator (2) and the denominator (3) by 20

STEP 3

Arrange the numerators in order being requested. In this question we are asked to arrange from the lowest to the highest

To convert the fraction to decimal write the numbers in the long division with 3 as the dividend and 8 as the divisor.

As 3 cannot be divided by 8, we can add 0 to 3 to make it 30 and add a decimal point in the quotient.

Continue the division, if you get a remainder that is smaller than 8, you can add 0 to it to make it divisible by 8.

Perform the division till you get the remainder 0 or till the remainder gets repeated.

Change the decimals to whole numbers with powers.

Count how many numbers are after the decimal points and express the whole number to 10 raise to the power of the negative of that number

0.5445

There are 4 numbers after the decimal points thus 5,4,4 and 5

0.5445 = 5445 x 10^-4

0.09

There are two numbers after the decimal points, thus 0 and 9

0.09 = 9 x 10^-2

9 goes into 5445, 605 times and itself once

Apply the law of indices to simplify the powers

Since base 10 in each of the indices, we can apply the low of indices

Thus if the bases are the same and dividing, you subtract the power of the denominator from the power of the numerator

NOTE: to change the result back to decimal, you move the decimal points as many times as the negative power to the left

Every whole number has a decimal point at the end. Hence 605 = 605.0

Since the power is negative 2 (-2), we move the decimal point to the left 2 times, landing after 6

Change the decimal to the form a x 10^-n

where a = whole number

n = how many times you have to move the decimal point to the right to get a whole number

0.725 = 725 x 10^-3

From the law of indices:

Cancel out

25 goes into 725, 29 times and 1000, 40 times

Step 1

Find the least common multiple (LCM)

LCM for 4, 3 and 12 is 12, thus

Multiples of 4

4,8,12,16,20,24,28,32,36,40,44,48

Multiples of 3

3,6,9,12,15,18,21,24,27,30,33,36

Multiples of 12

Multiples of 12 are numbers 12 can divide without a remainder (Simply the multiplication of 12). If you don't know the recitation of the multiples, simply add 12 to the preceding numbers to get the subsequent numbers starting from 12.

12,12+12=24,24+12 = 36,36+12 = 48,48+12 = 60,60+12 = 72,72+12 = 84,84+12 = 96,96+12 = 108,108+12 = 120,120+12 = 132,132+12 = 144,...

⇒ 12,24,36,48,60,72,84,96,108,120,132,144,...

From the above you can see that the least number which appeared in all the above multiples is 12, hence the LCM for 3,4 and 12 is 12

Step 2

Divide by the LCM and find out how many times the denominator goes into the LCM and multiply that by the numerator and maintain the signs

Thus for ¾ , the numerator is 3 and the denominator is 4 and 4(denominator) goes into 12(the LCM),3 times, hence 3 x 3 (numerator)

Thus for -⅓ , the numerator is 1 and the denominator is 3 and 3(denominator) goes into 12(the LCM),4 times, hence 4 x 1 (numerator) and maintain the negative sign.

Thus for +1⁄12 , the numerator is 1 and the denominator is 12 and 12(denominator) goes into 12(the LCM),1 time, hence 1 x 1 (numerator) and maintain the positive sign.

Now simplify the numerator first and afterward simplify the new fraction

As shown below

Fraction of a total is calculated as the value of what fraction is being calculated over the total

Thus fraction of a value = value/total, then you simplify the fraction.

Before you calculate the fraction, make sure they are of the same units. If one is in a different unit, change the other so they are all in the same unit.

Since the total (3 weeks) is in weeks, change the weeks to days

There are 7 days in 1 week hence 3 weeks will be 3 x 7 days = 21 days

Hence the fraction for the 18 days will be 18 days / 21 days

3 goes into 18, 6 times and into 21, 7 times. Hence the simplification of 18/21 = 6/7

Method I

Changing the fractions to descending and ordering.

$\frac{3}{4}$ = 0.75

$\frac{5}{8}$ = 0.625

The decimals will then be:

0.75, 0.8, 0.625, 0.65

Arranging the above in descending order will be:

0.8, 0.75, 0.65, 0.625

The descending order will therefore be:

0.8, $\frac{3}{4}$, 0.65, $\frac{5}{8}$

Method II

Changing the decimals to fractions and ordering

The fractions will be:

$\frac{3}{4}$, $\frac{4}{5}$, $\frac{5}{8}$, $\frac{13}{20}$

$\frac{3}{4}$ = $\frac{\mathrm{3\; x\; 10}}{\mathrm{4\; x\; 10}}$ = $\frac{30}{40}$

$\frac{4}{5}$ = $\frac{\mathrm{4\; x\; 8}}{\mathrm{5\; x\; 8}}$ = $\frac{32}{40}$

$\frac{5}{8}$ = $\frac{\mathrm{5\; x\; 5}}{\mathrm{8\; x\; 5}}$ = $\frac{25}{40}$

$\frac{13}{20}$ = $\frac{\mathrm{13\; x\; 2}}{\mathrm{20\; x\; 2}}$ = $\frac{26}{40}$

Now arrange them in descending order based on the magnitude of the numerators.

32 > 30 > 26 > 25

32 is for $\frac{4}{5}$ which was from 0.8

30 is for $\frac{3}{4}$

26 is for $\frac{13}{20}$ which was from 0.65

25 is for $\frac{5}{8}$

The descending order will therefore be:

0.8, $\frac{3}{4}$, 0.65, $\frac{5}{8}$

Fraction of the whole pocket money = 1

Fraction remains = 1 - Fraction spent

Fraction spent = Fraction spent on books + Fraction spent on pens

Fraction spent = $\frac{1}{4}$ + $\frac{1}{3}$

Fraction spent = $\frac{\mathrm{1\; x\; 3\; +\; 1\; x\; 4}}{12}$

Fraction spent = $\frac{\mathrm{3\; +\; 4}}{12}$

Fraction spent = $\frac{7}{12}$

Fraction remains = 1 - $\frac{7}{12}$

Every whole number is being divided by 1 but it is no written as a fraction with denominator 1

1 = $\frac{1}{1}$

Fraction remains = $\frac{1}{1}$ - $\frac{7}{12}$

Fraction remains = $\frac{\mathrm{1\; x\; 12\; -\; 7\; x\; 1}}{12}$

Fraction remains = $\frac{\mathrm{12\; -\; 7}}{12}$

Fraction remains = $\frac{5}{12}$