  Math Tests - mathematics practice questions

Which proportion is shaded (in fractions)? ## Fractions

In the real world, there are not only whole numbers. E.g. bread can be cut into two halves and pizza into eight slices. To describe them we use fractions. They simply tell us how many parts of a whole we have.

Fractions show parts of whole numbers. Each fraction consists of a numerator and a denominator. The numerator is the top digit that represents how many equal parts the fraction contains. The denominator is the bottom digit that represents how many equal parts the whole is broken into:

We can also consider fractions as numbers on a number line. In order to represent \$1/8\$ on the number line, draw the number line from 0 to 1 . Now, divide the gap between 0 and 1 into eight equal parts (denominator=8). This represents fractions from \$1/8\$ to \$8/8 (=1)\$.

Fraction on a number line ## Fractions

In the real world, there are not only whole numbers. E.g. bread can be cut into two halves and pizza into eight slices. To describe them we use fractions. They simply tell us how many parts of a whole we have.

Fractions show parts of whole numbers. Each fraction consists of a numerator and a denominator. The numerator is the top digit that represents how many equal parts the fraction contains. The denominator is the bottom digit that represents how many equal parts the whole is broken into:

We can also consider fractions as numbers on a number line. In order to represent \$1/8\$ on the number line, draw the number line from 0 to 1 . Now, divide the gap between 0 and 1 into eight equal parts (denominator=8). These represent fractions from \$1/8\$ to \$8/8 (=1)\$.

Compare fractions (with like numerators or denominators) ## Comparing fractions

A fraction is a part of a whole. To find the greater fraction you need to find the fraction that contains a greater part of the whole.

If fractions have the same denominator, you can compare numerators:

\$\$3/4 > 1/4\$\$

It is possible to draw it to see the result:

If there is a common numerator, the fraction with the greater denominator is actually smaller:

\$\$1/3 > 1/4\$\$

You can draw it as well:

If two fractions do not have common numerator or denominator, you can find equivalent fractions with the same denominator.
E.g. compare \$3/8\$ and \$1/3\$:

\$\${3×3}/{8×3}=9/24;;;{1×8}/{3×8}=8/24;→;\$\$
\$\$9/24>8/24\$\$

If you compare a fraction with a whole number, you can turn the whole number into a fraction with the same denominator:

\$\$1=1/1=2/2=3/3=4/4=5/5=...\$\$
\$\$4=8/2=12/3=16/4=20/5= ...\$\$

Compare 3 and \$9/4\$:

\$\$3=12/4;→;12/4>9/4\$\$

Compare fractions and whole numbers ## Fractions and whole numbers

If you compare a fraction with a whole number, you can turn the whole number into a fraction with the same denominator:

\$\$1=1/1=2/2=3/3=4/4=5/5=...\$\$
\$\$4=8/2=12/3=16/4=20/5= ...\$\$

Compare 3 and \$9/4\$:

\$\$3=12/4;→;12/4>9/4\$\$

Find the missing numerator or denominator (equivalent fractions) ## Equivalent fractions

Equivalent fractions are equal in value, even though they look different.

E.g.: \$;1/2=2/4\$
We can draw it:

When we multiply or divide the numerator (the top number) and the denominator (the bottom number) of a fraction by another fraction with the same numerator or denominator, it keeps the same value. A fraction with the same numerator and denominator is actually equal to 1 and by multiplying or dividing by 1 we won't change the value of original fraction.
E.g.:

\$\$2/3×2/2=4/6=4/6×2/2=8/12=\$\$
\$\$=8/12×3/3=24/36 ...\$\$
\$\$12/16={12÷4}/{16÷4}=3/4\$\$

If there is a missing part of a fraction in an equation, we need to find the equivalent fraction:

\$\$1/3=2/x\$\$

We need to multiply 1 by 2 to get 2. We need to do the same with the denominator (3). And 3×2=6

\$\$1/3=2/x;→;{1×2}/{3×2}\$\$
\$\$→;{3×2}=6;→;x=6\$\$

If there is a missing part of a fraction in an inequality, we need to find the equivalent fraction at first and then decrease/increase it to have it greater or smaller. E.g.:

\$\$3/8>6/x;→;{3×2}/{8×2}=6/16;→;3/8=6/16\$\$

Since \$3/8\$ are equivalent to \$6/16\$, we need to find a fraction that has 6 as a numerator (top number) and at the same time it is smaller than \$6/16\$. This is any fraction with a denominator greater than 16, e.g. \$6/17 ; 6/18 ; 6/19\$

\$\$x={17;;;18;;;19;;;20...}\$\$

Fractions of whole numbers ## Fractions of whole numbers

To find a fraction of a whole number (e.g. \$3/4\$ of 12), we multiply the numerator (3) by the given whole number (12) and then divide the product (36) by the denominator (4).

\$\${3×12}/4=36/4=9\$\$

Add and subtract fractions with like denominators (easy) If the denominators (the bottom numbers) are the same, we just need to add or subtract the numerators (the top numbers).

\$\$5/7+1/7=6/7\$\$
\$\$4/9−3/9=1/9\$\$

If the denominators (the bottom numbers) are different, we need to find the least common denominator. We can do this by following these steps:

\$\$1/4+1/2\$\$

Step 1: Convert fractions to have the same denominator (by multiplying):

\$\$1/4+1/2×2/2=1/4+2/4\$\$
Step 2: Add or subtract the numerators (the top numbers), and write the result over the same denominator (the bottom number).
\$\$1/4+2/4=3/4\$\$

Sometimes is better to divide one of the fractions:

\$\$4/6+2/3={4÷2}/{6÷2}+2/3=2/3+2/3=4/3\$\$

And in some cases you have to multiply or divide both fractions:

\$\$4/7−1/3=4/7×3/3−1/3×7/7=\$\$
\$\$=12/21−7/21=5/21 \$\$

If you are adding/subtracting fractions and whole numbers, you can turn the whole number into a fraction with the same denominator.

\$\$1/5+1=?\$\$
\$\$1=1/1=2/2=3/3=4/4=5/5\$\$
\$\$;→;1/5+5/5=6/5\$\$
\$\$13/3−4=?\$\$
\$\$4=4/1=8/2=12/3;→;13/3−12/3=1/3\$\$

Add and subtract fractions with unlike denominators (medium) If the denominators (the bottom numbers) are the same, we just need to add or subtract the numerators (the top numbers).

\$\$5/7+1/7=6/7\$\$
\$\$4/9−3/9=1/9\$\$

If the denominators (the bottom numbers) are different, we need to find the least common denominator. We can do this by following these steps:

\$\$1/4+1/2\$\$

Step 1: Convert fractions to have the same denominator (by multiplying):

\$\$1/4+1/2×2/2=1/4+2/4\$\$
Step 2: Add or subtract the numerators (the top numbers), and write the result over the same denominator (the bottom number).
\$\$1/4+2/4=3/4\$\$

Sometimes is better to divide one of the fractions:

\$\$4/6+2/3={4÷2}/{6÷2}+2/3=2/3+2/3=4/3\$\$

And in some cases you have to multiply or divide both fractions:

\$\$4/7−1/3=4/7×3/3−1/3×7/7=\$\$
\$\$=12/21−7/21=5/21 \$\$

If you are adding/subtracting fractions and whole numbers, you can turn the whole number into a fraction with the same denominator.

\$\$1/5+1=?\$\$
\$\$1=1/1=2/2=3/3=4/4=5/5\$\$
\$\$;→;1/5+5/5=6/5\$\$
\$\$13/3−4=?\$\$
\$\$4=4/1=8/2=12/3;→;13/3−12/3=1/3\$\$

Mixed numbers ## Mixed numbers

A proper fraction is a fraction with the numerator (top number) less than the denominator (bottom number), e.g. \$1/3 ;;; 3/4 ;;; 5/7 ;;; 6/7\$. It is always less than 1.

An improper fraction is a fraction with the numerator larger than (or equal to) the denominator, e.g. \$7/6 ;;; 8/4 ;;; 3/3 ;;; 8/7 ;;; 34/9\$. It is always greater than or equal to 1.

A mixed number is a whole number and a proper fraction combined. Thus, it is always greater than 1.

Each improper fraction can also be expressed as a mixed number.

How to convert an improper fraction to a mixed number.

\$\$15/7\$\$

To convert \$15/7\$ into a mixed number, we need to find out how many times 7 can fit into 15. For this divide the numerator by the denominator:
\$15÷7=2\$ with remainder 1 (\$15=2×7+1\$).
2 will be used as the whole number of mixed number and 1 will be the numerator above the denominator.

\$\$15/7=2{1/7}\$\$

How to convert a mixed number into a fraction:

\$\$5{3/4}=?\$\$

Multiply the whole number by the denominator:

\$\$5×4=20\$\$
\$\$20+3=23\$\$
Write 23 over the denominator
\$\$5{3/4}=23/4\$\$

A mixed number can be also written as: 158 ; 334.

Dividing fractions ## Dividing fractions

To divide two fractions, find the reciprocal of the second fraction (invert it) and multiply.

\$\${8/9}÷{2/7}={8/9}×{7/2}=56/18\$\$

You can simplify/reduce the resulting fraction (or convert to a mixed number):

\$\${56÷2}/{18÷2}=28/9={3×9+1}/9=3{1/9}\$\$
Dividing fractions by a whole number

Rewrite the whole number as a fraction (just put the whole number over 1 as denominator):

\$\${4/5}÷7={4/5}÷{7/1}=4/5×1/7=4/35\$\$

Complex fractions ## Complex fractions

In general, a complex fraction is normally formed of two fractions, one on top of the other (the numerator, denominator, or both contain a fraction). It is actually a ratio of two fractions.

To convert this to a simple fraction, rewrite as division:

\$\${;4/5;}/{;2/3;}={4/5}÷{2/3}={4/5}×{3/2}=12/10\$\$

You can also just multiply the top numerator by the bottom denominator and the top denominator by the bottom numerator:

\$\${;4/5;}/{;2/3;}={4×3}/{5×2}=12/10\$\$

If only the numerator or the denominator contains a fraction, rewrite the other one as a fraction, and then divide or multiply:

\$\${;1/8;}/3={;1/8;}/{;3/1;}={1×1}/{8×3}=1/24\$\$

\$\${;7;}/{4/5}={{;7/1;}/{;4/5;}={7×5}/{1×4}=35/4\$\$