smc-4608-15-Simplifying ratios

Ratio, SM-Bank 039 MC

A Christmas pudding recipe uses 3 cups of sugar for every 4 cups of sultanas.

Select the correct combination of sugar and sultanas for this recipe.

$\dfrac{1}{3}\ \text{cup of sugar, }\dfrac{1}{4}\ \text{cup of sultanas}$
$1\ \text{cup of sugar, }1.5\ \text{cups of sultanas}$
$1.5\ \text{cups of sugar, }2\ \text{cups of sultanas}$
$4\ \text{cups of sugar, }5\ \text{cups of sultanas}$

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$C$

Show Worked Solution

$\text{Ratio}$	$= 3:4$
	$=1.5:2$

$\therefore\ 1.5\ \text{cups of sugar, }2\text{ cups of sultanas}$

$\text{is in the correct ratio.}$

$\Rightarrow C$

Ratio, SM-Bank 016

Maude has 55 cents in coins and Will has $2.70 in coins.

Write Maude's money as a ratio of Will's money in cents. (1 mark)

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Express the ratio in simplest form. (2 marks)

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a. $55:270$

b. $11:54$

Show Worked Solution

$55\ \text{cents to}\ $2.70$

$=55:270$

b.	$55:270$	$=\dfrac{55}{5}:\dfrac{270}{5}$
		$=11:54$

Ratio, SM-Bank 015

Change these ratios to the same units and then write in simplest form.

$40\ \text{cm}: 1.2\ \text{m}$ (2 marks)

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$4\ \text{hours}:20\ \text{minutes}$ (2 marks)

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$15\ \text{days}: 3\ \text{weeks}$ (2 marks)

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a. $1:3$

b. $6:1$

c. $5:7$

Show Worked Solution

a.	$40\ \text{cm}:1.2\ \text{m}$	$=40\ \text{cm}:120\ \text{cm}$
		$=40:120$
		$=1:3$

b.	$4\ \text{hours}:20\ \text{minutes}$	$=240\ \text{minutes}:40\ \text{minutes}$
		$=240:40$
		$=6:1$

c.	$15\ \text{days}:3\ \text{weeks}$	$=15\ \text{days}:21\ \text{days}$
		$=15:21$
		$=5:7$

Ratio, SM-Bank 014

Fully simplify the following ratios.

$\dfrac{5}{12}:\dfrac{11}{8}$ (2 mark)

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$1\dfrac{1}{4}:3\dfrac{1}{2}$ (2 marks)

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a. $10:33$

b. $5:14$

Show Worked Solution

a.	$\dfrac{5}{12}:\dfrac{11}{8}$	$=\bigg(\dfrac{5}{12}\times 24\bigg):\bigg(\dfrac{11}{8}\times 24\bigg)$	$\ \ \text{Multiply by LCM of 12 and 8.}$
		$=10:33$

b.	$1\dfrac{1}{4}:3\dfrac{1}{2}$	$=\dfrac{5}{4}:\dfrac{7}{2} \ \ \ \text{Convert to improper fractions }$
		$=\bigg(\dfrac{5}{4}\times 4\bigg):\bigg(\dfrac{7}{2}\times 4\bigg)\ \ \ \ \text{Multiply by LCM of 4 and 2. }$
		$=5:14$

Ratio, SM-Bank 013

Fully simplify the following ratios.

$\dfrac{1}{2}:3$ (1 mark)

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$\dfrac{1}{4}:\dfrac{1}{3}$ (2 marks)

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a. $1:6$

b. $3:4$

Show Worked Solution

a.	$\dfrac{1}{2}:3$	$=\Bigg(\dfrac{1}{2}\times 2\Bigg):\Bigg(3\times 2\Bigg)$	$\ \ \text{Multiply by LCM of 2 and 1.}$
		$=1:6$

b.	$\dfrac{1}{4}:\dfrac{1}{3}$	$=\Bigg(\dfrac{1}{4}\times 12\Bigg):\Bigg(\dfrac{1}{3}\times 12\Bigg)$	$\ \ \text{Multiply by LCM of 3 and 4. }$
		$=3:4$

Ratio, SM-Bank 012

Fully simplify the following ratios.

$10:32$ (1 mark)

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$144:60$ (1 mark)

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$36:18:66$ (1 mark)

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a. $5:16$

b. $12:5$

c. $6:3:11$

Show Worked Solution

a.	$10:32$	$=\dfrac{10}{2}:\dfrac{32}{2}$	$\ \ \text{(Divide by HCF of 10 and 32.)}$
		$=5:16$

b.	$144:60$	$=\dfrac{144}{12}:\dfrac{60}{12}$	$\ \ \text{(Divide by HCF of 144 and 60.)}$
		$=12:5$

c.	$36:18:66$	$=\dfrac{36}{6}:\dfrac{18}{6}:\dfrac{66}{6}$	$\ \ \text{(Divide by HCF of 36, 18 and 66.) }$
		$=6:3:11$

Ratio, SM-Bank 011

Fully simplify the following ratios.

$24:18$ (1 mark)

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$180:81$ (1 mark)

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$14:49:56$ (1 mark)

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a. $4:3$

b. $20:9$

c. $2:7:8$

Show Worked Solution

a.	$24 :18$	$=\dfrac{24}{6}:\dfrac{18}{6}$	$\ \ \text{(Divide by HCF of 24 and 18.)}$
		$=4:3$

b.	$180 :81$	$=\dfrac{180}{9}:\dfrac{81}{9}$	$\ \ \text{(Divide by HCF of 180 and 81.)}$
		$=20:9$

c.	$14 :49:56$	$=\dfrac{14}{7}:\dfrac{49}{7}:\dfrac{56}{7}$	$\ \ \text{(Divide by HCF of 14, 49 and 56.)}$
		$=2:7:8$

Ratio, SM-Bank 010 MC

Which of the following ratios is not in simplest form?

$25:4$
$9:63$
$108:11$
$6:35$

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$B$

Show Worked Solution

$9 :63$	$=\dfrac{9}{9}:\dfrac{63}{9}$	$\ \ \text{Divide by HCF of 9 and 63. }$
	$=1:7$

$\Rightarrow B$

Ratio, SM-Bank 009 MC

Which of the following ratios is not in simplest form?

$4:7$
$13:3$
$1:0.5$
$2:9$

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$C$

Show Worked Solution

$1 :0.5$	$=1:\dfrac{1}{2}$	$\ \ \ \text{(Multiply by 2 to remove fraction.)}$
	$=2:1$

$\Rightarrow C$

Ratio, SM-Bank 008

Complete each pair of equivalent ratios. (3 marks)

a.	$16:12=$	$:3$
b.	$18:24=$	$:4$
c.	$54:9=$	$:1$

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a. $4:3$

b. $3:4$

c. $6:1$

Show Worked Solution

a. $16 :12$	$=(16÷4):(12÷4)$
	$=4:3$

b. $18 :24$	$=(18÷6):(24÷6)$
	$=3:4$

c. $54 :9$	$=(54÷9):(9÷9)$
	$=6:1$

Ratio, SM-Bank 007

Complete each pair of equivalent ratios. (3 marks)

a. $3:4$	$=9:$
b. $2:5$	$=8:$
c. $10:3$	$=30:$

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a. $9:12$

b. $8:20$

c. $30:9$

Show Worked Solution

a. $3 :4$	$=3\times 3:4\times 3$
	$=9:12$

b. $2 :5$	$=2\times 4:5\times 4$
	$=8:20$

c. $10 :3$	$=10\times 3:3\times 3$
	$=30:9$

Ratio, SM-Bank 006 MC

At the Year 7 camp, there were 20 teachers and 180 students.

What is the ratio of students to the total number of people at the camp?

$9:2$
$180:20$
$9:10$
$9:1$

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$C$

Show Worked Solution

$\text{students}\ :\ \text{total}$	$=180:(20+180)$
	$=180:200$
	$=9:10$

$\Rightarrow C$

Ratio, SM-Bank 005 MC

In a boarding kennel, there are 15 dogs and 10 cats.

What is the ratio of cats to dogs?

$2:3$
$15:10$
$2:5$
$3:5$

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$A$

Show Worked Solution

$\text{cats}\ :\ \text{dogs}$	$=10:15$
	$=2:3$

$\Rightarrow A$

Ratio, SM-Bank 004 MC

In a hospital, there are 40 nurses and 600 patients.

What is the ratio of nurses to patients?

$2:3$
$1:15$
$30:2$
$1:16$

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$B$

Show Worked Solution

$\text{Nurses : Patients}$

$=40:600=1:15$

$\Rightarrow B$

b.	\(55:270\)	\(=\dfrac{55}{5}:\dfrac{270}{5}\)
		\(=11:54\)

a.	\(40\ \text{cm}:1.2\ \text{m}\)	\(=40\ \text{cm}:120\ \text{cm}\)
		\(=40:120\)
		\(=1:3\)

b.	\(4\ \text{hours}:20\ \text{minutes}\)	\(=240\ \text{minutes}:40\ \text{minutes}\)
		\(=240:40\)
		\(=6:1\)

c.	\(15\ \text{days}:3\ \text{weeks}\)	\(=15\ \text{days}:21\ \text{days}\)
		\(=15:21\)
		\(=5:7\)

a.	\(\dfrac{5}{12}:\dfrac{11}{8}\)	\(=\bigg(\dfrac{5}{12}\times 24\bigg):\bigg(\dfrac{11}{8}\times 24\bigg)\)	\(\ \ \text{Multiply by LCM of 12 and 8.}\)
		\(=10:33\)

\(\text{Ratio}\)	\(= 3:4\)
	\(=1.5:2\)

a.	\(\dfrac{1}{2}:3\)	\(=\Bigg(\dfrac{1}{2}\times 2\Bigg):\Bigg(3\times 2\Bigg)\)	\(\ \ \text{Multiply by LCM of 2 and 1.}\)
		\(=1:6\)

b.	\(\dfrac{1}{4}:\dfrac{1}{3}\)	\(=\Bigg(\dfrac{1}{4}\times 12\Bigg):\Bigg(\dfrac{1}{3}\times 12\Bigg)\)	\(\ \ \text{Multiply by LCM of 3 and 4. }\)
		\(=3:4\)

a.	\(10:32\)	\(=\dfrac{10}{2}:\dfrac{32}{2}\)	\(\ \ \text{(Divide by HCF of 10 and 32.)}\)
		\(=5:16\)