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The pupil notices that he cannot reduce to an exact decimal fraction, because he always has the remainder 4.

Give enough examples like the three above to fix the idea firmly with the pupil.

Reducing Mixed Numbers to Decimals.

There is nothing new for the pupil here. He need only be told to take the fractional part of the number and divide as in the preceding case. Then add the decimal fraction found to the whole numberthe integer.

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The pupil has already discovered that some commom fractions cannot be changed to exact decimal fractions, as

1.33333 on to infinity.
= .66666 on to infinity.
3.212121, etc.

These decimals are known as Circulates, Recurring or Circulating decimals.

The part which recurs is called the Repetend.

This is marked by putting a dot over the first and last figures of it. For instance, if we write the 21 in the last case above, this way: 21, it indicates that, if written out, the result would be 21212121, etc., on to infinity.

Where a circulating decimal occurs in work, it is best to reduce it to a common fraction. If need be, it may be expressed in the result, as a circulate to any number of decimal places.



Since problems involving decimals have but one new feature, the use of the decimal point, there is no different reasoning process presented in them from that in problems involving the fundamental operations.

In consequence, the only difficulty that may present itself, if the early work has been properly done, is with reference to the use of the decimal point.

For illustration and practice a few problems are, however, added.


A city has parks with these areas: 4.36 A., 7.5 A., 18.625 A., and 7.03 A. What is the total area of its

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inches. What is the

The length of a second's pendulum is 39.1392 inches and a meter is 39.371 difference in their lengths?






The problem is one in subtraction. Inspection shows by the first decimal figure that the meter is the longer,

so we write it for the minuend. Write with the decimal points in a column, and subtract as in integers.


A merchant sold cloth which cost him 3.5g a yard, for 5. What was his profit on 350 yds?

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In 1900 the population of all Alaska was 63,592. The population of Chicago at the same time was 1,698,575. Find in a decimal fraction to four places, the part the population of Alaska was of the population of Chicago.

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A decimal fraction is a fraction whose denominator is ten or some power of ten.

A decimal point is a dot used to locate units.

A decimal is a decimal fraction whose denominator is shown by the number of places it ends to the right of the decimal point. Only the numerator is written.

A complex decimal is a decimal with a commor fraction at the right, as, .121.

A mixed decimal is a whole number with a decimal fraction to its right, as, 34.5.


Write the numerator. From the right, point off as many decimal places as there are O's in the denomi nator. Prefix 0's to the numerator, if needed.


Read the numerator as a simple number and call the whole by the name of the right-hand order. To reduce decimals to a common denominator, Annex O's until their decimal places are equal.

To reduce common fractions to decimals,

Annex decimal O's to the right of the numerator and divide by the denominator.

To reduce a decimal to a common fraction,

Write the figures of the decimal as a numerator, with 1 and as many O's as places in the decimal, for the denominator. Reduce to lowest terms.


1. Annexing O's to the right of a decimal, or removing them therefrom, does not change its value.

2. Taking O's from the left of a decimal increases its value tenfold for each 0 removed.

3. Prefixing O's to a decimal diminishes its value to one tenth for each 0 prefixed


Business men often mark goods in such a way that only themselves and their clerks know the cost and selling price. To do this, they use some word or phrase containing ten letters, each representing a figure, as


1 2 3 4 5 6 7 8 9 0

If the cost of a pair of shoes, for instance, is $2.10, the merchant marks them mie. If he wishes to sell them for $3.25, he might mark the selling price pmr. For quick comparison they would be written on the card attached to the goods in this way:



To prevent the use of the same letter twice in succession, an extra letter, or repeater, is used. Any letter not found in the key may be used as a repeater.

If the cost price were $4.40, it could be written ode, using d as the repeater.

Any word or phrase containing ten letters, no two of which are the same, may be used as a key. A few examples are:




For the Pupil to Do:

1. Write the cost and selling price for several articles, using one of the above as a key.

2. Invent a new key.

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