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Ratio word problems (difficult) ## Ratios

A ratio (proportion) expresses the relationship of one quantity to another. Ratios are usually used to compare two or multiple numbers/quantities, e.g. 1:2 or 4:1:3.

The ratio of two quantities can be written as a fraction, e.g. 1:2 can be written as \$1/2\$. It is read as “1 to 2”. It means that the second quantity is twice as large as the first.

The numbers 1 and 2 are called terms (antecedent:consequent). The order is important. The ratio 1:2 is not the same as 2:1. Ratios can be simplified/reduced by dividing it by the same number. Ratio 15:3 is the same as 5:1 (We have divided both terms by 3.) We can read it as 15 is to 3 as 5 is to 1. We call them equivalent ratios (they have the same value, even though they look different). A ratio expressed in lowest terms is a ratio by which both terms can’t be divided further by a common divisor (1:8:9 or 2:3 or 4:5:7)

A ratio can describe a part-to-part comparison or a part-to-whole comparison.

Let’s have 12 girls and 4 boys in a class (altogether 16 children). Their part-to-part ratio (girls to boys) is 12:4 (can be simplified to 3:1). Thus, ratio of boys to girls is 1:3. And ratio of girls to all children (part-to-whole) is 12:16 (or 3:4 after simplifying).

The ratio actually only expresses parts of a certain whole to us. If the ratio of girls to boys is 3: 1, it means that the girls are \$3/4\$ of it and boys \$1/4\$.

### Dividing a number into two (or more) parts in a given ratio

Let the number be 20. It is to be divided into two parts in the ratio 3:7 . The two parts of 20 we are looking for are x and y. x:y will be equivalent to 3:7 and at the same time x+y=20

\$\$x/y=3/7=k\$\$

Number k is a number with which we will multiply at the end to transform the relative part of the whole to absolute quantity. If we split 20 to 10 parts (because 3+7=10) it corresponds with 1 part of it.

Let’s find x, y:

\$\$ x = 3k ; y = 7k\$\$
\$\$3k+7k=20 ;→;k=20/{3+7}\$\$
\$\$k=2 ;→;x=3k=6;;; y=7k=14\$\$

In other words, if a ratio is a:b we have to add up ratio terms (a+b or 3+7=10) and use this sum to divide original number (20) by it to get k: 20÷10=2. Then we have to multiply k with terms of the ratio to get the results 3k=6; 7k=14. Their sum is equal to the original number (6+14=20)

### Part-to-whole word problems

The ratio of girls to all students in a school is 3:7. There is 1400 students altogether. How many boys are there?

1400 students should be divided by 7 to get k.

\$\$1400÷7=200=k\$\$

Number of girls is 3×k=3×200=600. The rest will be boys (1400−600=800).

Another way is to set a ratio part-to-whole for boys and that’s (7−3):7=4:7 . With k equal to 200, number of boys is 4×k=4×200=800.