Ratios (HSC SSCE Mathematics Standard): Revision Notes

Worked Example: Simplify $20:4$

• Identify the highest common factor of $20$ and $4$, which is $4$

• Divide both parts by $4$:

  $20 \div 4 : 4 \div 4 = 5:1$

The simplified ratio is $5:1$, meaning for every $5$ parts of the first quantity, there is $1$ part of the second quantity.

Worked Example: Simplify $3:\frac{1}{2}$

• The lowest common denominator is $2$ (since we only have halves)

• Multiply both parts by $2$:

  $3 \times 2 : \frac{1}{2} \times 2$

• Calculate the result:

  $6:1$

The simplified ratio is $6:1$. This tells us there are $6$ parts of the first quantity for every $1$ part of the second quantity.

Worked Example: Simplify $1.5:3.5$

• Multiply both parts by $10$ to remove decimals:

  $1.5 \times 10 : 3.5 \times 10 = 15:35$

• Find the highest common factor of $15$ and $35$, which is $5$

• Divide both parts by $5$:

  $\frac{15}{5} : \frac{35}{5} = 3:7$

The simplified ratio is $3:7$.

Worked Example: Mikhail and Ilya were given $450 to share in the ratio $4:5$. How much did each person receive?

Step 1: Calculate total parts

• Total parts $= 4 + 5 = 9$

Step 2: Find the value of one part

• $9$ parts $= 450$

• $1$ part $= 450 \div 9 = 50$

Step 3: Calculate each share

• Mikhail's share: $4$ parts $= 4 \times 50 = 200$

• Ilya's share: $5$ parts $= 5 \times 50 = 250$

Step 4: Check the answer

• $200 + 250 = 450$ ✓

Therefore, Mikhail received $200 and Ilya received $250.

Worked Example: An inheritance of $6000 is to be divided among three children, Xia, Yui and Zi, in the ratio $5:8:7$. Calculate each child's share.

Step 1: Calculate total parts

• Total parts $= 5 + 8 + 7 = 20$

Step 2: Find the value of one part

• $20$ parts $= 6000$

• $1$ part $= 6000 \div 20 = 300$

Step 3: Calculate each share

• Xia's share: $5$ parts $= 5 \times 300 = 1500$

• Yui's share: $8$ parts $= 8 \times 300 = 2400$

• Zi's share: $7$ parts $= 7 \times 300 = 2100$

Step 4: Check the answer

• $1500 + 2400 + 2100 = 6000$ ✓

Therefore, Xia received $1500, Yui received $2400, and Zi received $2100.