Convert Between Forms and Compute Efficiently | Grade 7 Math | LevelKite

When you solved two-step equations, each step gave you a simpler equation with the same solution. This lesson does something similar with numbers: you rewrite them so they are easier to work with, without changing their value.

A rational number is a number you can write as a fraction of two integers, where the denominator is not zero. You can write the same rational number in different forms: as a fraction, as a decimal, or as a percent of a stated positive amount. For example, $\frac{1}{4}$ of $48$ , $0.25$ of $48$ and $25\%$ of $48$ are three ways to name the same amount.

Why switch forms? Because one form is often easier to compute with. Finding $25\%$ of $48$ looks like a chore. Finding $\frac{1}{4}$ of $48$ is just $48 \div 4 = 12$ . It is the same question in a friendlier form.

In this lesson you will convert numbers, which means rewriting them in another form. Then you will use facts about addition and multiplication to reorder and regroup a calculation.

Percent means per hundred. A percent is a rate per $100$ of a stated positive whole, the full amount it is measured against. It tells you how many parts you have for every $100$ equal parts of that whole. So $30\%$ of a whole is $30$ of its $100$ equal parts, or $\frac{30}{100}$ of it. That meaning gives you every conversion.

Percent to fraction or decimal: write the number over $100$ , because that is what the percent sign means. $30\% = \frac{30}{100} = \frac{3}{10} = 0.3$ .
Decimal to percent: multiply by $100$ and write the result over $100$ . That multiplies by $\frac{100}{100} = 1$ , so the value stays the same. $0.45 = \frac{0.45 \times 100}{100} = \frac{45}{100} = 45\%$ .
Fraction to decimal: a fraction bar means division, so divide the numerator by the denominator, which is never zero. $\frac{3}{8} = 3 \div 8 = 0.375$ (check: $8 \times 0.375 = 3$ ).
Fraction to percent: convert to a decimal first, then to a percent. $\frac{3}{8} = 0.375 = \frac{37.5}{100} = 37.5\%$ .

Each conversion rewrites the same number, so its value never changes.

The same number in three forms. Each percent is a rate per 100 of one stated positive whole.

Fraction	Decimal	Percent
$\frac{1}{2}$	$0.5$	$50\%$
$\frac{1}{4}$	$0.25$	$25\%$
$\frac{3}{4}$	$0.75$	$75\%$
$\frac{1}{5}$	$0.2$	$20\%$
$\frac{1}{10}$	$0.1$	$10\%$
$\frac{1}{20}$	$0.05$	$5\%$
$\frac{1}{8}$	$0.125$	$12.5\%$
$\frac{1}{3}$	$0.333\ldots$ (the $3$ repeats forever)	$33\frac{1}{3}\%$

Which form should you use? Let the other numbers in the problem decide.

Use the fraction when its denominator divides the other number evenly. $12.5\%$ of $72$ is $\frac{1}{8}$ of $72$ , and $72 \div 8 = 9$ .
Use the decimal when the other numbers are decimals, such as money, so the place values line up. Three fourths of a dollar added to $2.50$ dollars is $0.75 + 2.50 = 3.25$ dollars.
Use the fraction when the decimal never ends. $\frac{1}{3}$ of $27$ is exactly $27 \div 3 = 9$ , but the rounded decimal gives $0.33 \times 27 = 8.91$ , which is close but not exact.
Use the percent when the question asks for one, and say which positive whole it is a percent of.

Worked example

Convert a percent to a fraction

A jacket has an original price of $45$ dollars. A store sells it at $20\%$ off the original price. What is the sale price?

Name the whole. The discount is 20% of the original price, so the whole is the original 45 dollars, a stated positive amount.
$\text{discount} = 20\% \text{ of } 45 \text{ dollars}$
Convert the percent to a fraction. Percent means per hundred, so write 20 over 100. Then divide the numerator and the denominator by 20; dividing both by the same nonzero number keeps the value. The fraction is the handy form here because 45 divides evenly by 5.
$20\% = \frac{20}{100} = \frac{20 \div 20}{100 \div 20} = \frac{1}{5}$
Find the discount. Taking one fifth of 45 means splitting 45 into 5 equal parts and keeping one part.
$\frac{1}{5} \times 45 = 45 \div 5 = 9$
Subtract the discount from the original price, because the sale price is what remains after the discount is taken off.
$45 - 9 = 36$

Answer

The sale price of the jacket is $36$ dollars.

The second tool changes the arrangement of a calculation, not the form of a number. A property of operations is a fact about addition or multiplication that is true for all rational numbers, whether positive, negative or zero. The table lists three of them, where $a$ , $b$ and $c$ stand for any rational numbers. The first two, the commutative and associative properties, say that order and grouping do not matter when you only add, or only multiply. The third, the distributive property, says that multiplying a sum gives the same result as multiplying each part and then adding: $12$ groups of $4.5$ are $12$ groups of $4$ together with $12$ groups of $0.5$ .

Three properties of operations for rational numbers

Property	What it lets you do	In symbols, for any rational a, b, c	Example
Commutative property	Change the order of numbers you add, or of numbers you multiply	$a + b = b + a$ and $a \times b = b \times a$	$-2.5 + 6 = 6 + (-2.5) = 3.5$
Associative property	Change the grouping of numbers you add, or of numbers you multiply	$(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$	$(17 \times 0.25) \times 4 = 17 \times (0.25 \times 4) = 17 \times 1 = 17$
Distributive property	Multiply a sum one part at a time, or pull out a number that multiplies every part	$a \times (b + c) = a \times b + a \times c$	$12 \times 4.5 = 12 \times (4 + 0.5) = 12 \times 4 + 12 \times 0.5 = 48 + 6 = 54$

The commutative and associative properties are about addition and multiplication only. Changing the order in a subtraction can change the result: $8 - 3 = 5$ , but $3 - 8 = -5$ . The same is true for division: $8 \div 2 = 4$ , but $2 \div 8 = 0.25$ .

So before you reorder or regroup a subtraction, rewrite it as adding the opposite. The opposite of $b$ is $-b$ , the number that adds with $b$ to make $0$ , and $a - b = a + (-b)$ for all rational numbers $a$ and $b$ . For example, $8 - 3 = 8 + (-3) = (-3) + 8 = 5$ .

Key concept

Change the form or the arrangement, never the value

Converting rewrites a number as a fraction, a decimal or a percent without changing its value. The commutative, associative and distributive properties rearrange additions and multiplications of rational numbers without changing the result. So before you compute, look for the form and the arrangement that make the arithmetic easy.

Two conditions keep this safe. A percent is always a rate per $100$ of a stated positive whole. And a subtraction must be rewritten as adding the opposite before you reorder or regroup it.

Worked example

Reorder and regroup to make easy sums

Use properties of operations to compute $6.4 - 3.75 + 3.6 - 1.25$ efficiently.

Rewrite each subtraction as adding the opposite. Now every operation is an addition, so the commutative and associative properties apply.
$6.4 + (-3.75) + 3.6 + (-1.25)$
Use the commutative property of addition to change the order. Put 6.4 next to 3.6 because their decimal parts add to exactly 1 (0.4 + 0.6 = 1). Put the two negative numbers together because their decimal parts also add to exactly 1 (0.75 + 0.25 = 1).
$6.4 + 3.6 + (-3.75) + (-1.25)$
Use the associative property of addition to group each pair, then add inside each group. Two negative numbers add to a negative number: 3.75 + 1.25 = 5, so the second group equals negative 5.
$(6.4 + 3.6) + [(-3.75) + (-1.25)] = 10 + (-5)$
Add the two results. Adding negative 5 is the same as subtracting 5.
$10 + (-5) = 10 - 5 = 5$
Check by working left to right in the original order. The result matches, so reordering and regrouping did not change the value.
$6.4 - 3.75 = 2.65, \quad 2.65 + 3.6 = 6.25, \quad 6.25 - 1.25 = 5$

Answer

The value of $6.4 - 3.75 + 3.6 - 1.25$ is $5$ .

Worked example

Convert twice and use the distributive property

A meal costs $32$ dollars. Sales tax is $5\%$ of the meal cost, and you leave a tip that is $20\%$ of the meal cost. What is the total amount you pay?

Name the whole and convert. Both percents are rates per 100 of the same stated whole, the meal cost of 32 dollars, which is positive. Write each percent over 100. Take care: 5% is 0.05, not 0.5, because 0.5 is 50 hundredths, which is 50%.
$5\% = \frac{5}{100} = 0.05, \qquad 20\% = \frac{20}{100} = 0.20$
Write the tax plus the tip. A part of an amount is found by multiplying the amount by the fraction or its equal decimal, so each decimal multiplies the meal cost of 32.
$32 \times 0.05 + 32 \times 0.20$
Both parts multiply the same number, 32. Use the distributive property to add the decimals first, so you multiply only once.
$32 \times 0.05 + 32 \times 0.20 = 32 \times (0.05 + 0.20) = 32 \times 0.25$
Convert 0.25 to a fraction: it is 25 hundredths, which simplifies to one fourth when you divide 25 and 100 by 25. The fraction is handy because 32 divides evenly by 4. The result is the tax and tip together, in dollars.
$0.25 = \frac{25}{100} = \frac{1}{4}, \qquad 32 \times \frac{1}{4} = 32 \div 4 = 8$
Add the tax and tip to the meal cost to get the total.
$32 + 8 = 40$
Check by finding each part separately. The total matches.
$32 \times 0.05 = 1.60, \quad 32 \times 0.20 = 6.40, \quad 32 + 1.60 + 6.40 = 40$

Answer

You pay $40$ dollars in total: $32$ dollars for the meal plus $8$ dollars of tax and tip.

Takeaway: before you compute, pause and look. Can a number change form, such as a percent of a stated positive whole becoming a fraction, so the arithmetic gets easier? Can you reorder or regroup the additions and multiplications so the numbers combine easily, after rewriting any subtraction as adding the opposite? Both moves change how a problem looks and never what it equals.

Next, you will use these tools to solve longer multi-step problems with rational numbers in any form, and you will estimate to judge whether an answer is reasonable.