Ratio Problems: Additional Practice (Set 2)

Write the ages as $4k$ and $3k$. In $6$ years, $(4k + 6) : (3k + 6)$ equals $6 : 5$, so cross-multiply.

$5(4k + 6) = 6(3k + 6) \implies 20k + 30 = 18k + 36 \implies k = 3$

Anna is $4k = 12$.

Sheep minus cows is $9k - 4k = 5k$, so $5k = 65$ and $k = 13$. The total is all $9 + 4 = 13$ parts.

$5k = 65 \implies k = 13, \quad 13k = 169$

There are $169$ animals.

Flour is the $5$ term, so $5k = 15$ and $k = 3$. The whole mixture is all $2 + 5 + 1 = 8$ parts.

$5k = 15 \implies k = 3, \quad 8k = 24$

The recipe makes $24$ cups.

Scale $y$ to the least common multiple of $2$ and $6$, which is $6$: $5 : 2 = 15 : 6$, so $x : y : z = 15 : 6 : 7$. Then $z = 7k = 42$, so $k = 6$.

$7k = 42 \implies k = 6, \quad 15k = 90$

So $x = 90$.

Write almonds as $5k$ and cashews as $3k$. Removing $8$ almonds makes them equal, so $5k - 8 = 3k$.

$5k - 8 = 3k \implies 2k = 8 \implies k = 4$

Cashews is $3k = 12$.

Write length as $7k$ and width as $4k$. The perimeter is $2(7k + 4k) = 22k = 88$, so $k = 4$.

$22k = 88 \implies k = 4, \quad 7k = 28$

The length is $28$ cm.

Write the numbers as $5k$ and $2k$. Adding $9$ to the smaller matches the larger, so $2k + 9 = 5k$.

$2k + 9 = 5k \implies 3k = 9 \implies k = 3$

The larger number is $5k = 15$.

Write the numbers as $5k$ and $3k$. After adding $10$ to each, $(5k + 10) : (3k + 10)$ equals $7 : 5$, so cross-multiply.

$5(5k + 10) = 7(3k + 10) \implies 25k + 50 = 21k + 70 \implies k = 5$

The larger original number is $5k = 25$.

Let the third get $1$ part. The second is three times that, $3$ parts, and the first is twice the second, $6$ parts, so the ratio is $6 : 3 : 1$. The parts total $10$, so $10k = 7200$ and $k = 720$.

$10k = 7200 \implies k = 720, \quad 1k = 720$

The third person gets $720$ dollars.

The shared term $B$ is $5$ in both, and $C : B = 3 : 5$ makes $C = 3$ when $B = 5$, so $A : B : C = 6 : 5 : 3$.

$A : C = 6 : 3 = 2 : 1$

Write antifreeze as $3k$ and water as $7k$. The water is unchanged, so $(3k + 16) : 7k$ equals $5 : 7$, and cross-multiplying gives the equation in $k$.

$7(3k + 16) = 5(7k) \implies 21k + 112 = 35k \implies k = 8$

Water is $7k = 56$ liters.

Smallest plus largest is $4k + 9k = 13k$, so $13k = 52$ and $k = 4$. The middle number is the $6$ term.

$13k = 52 \implies k = 4, \quad 6k = 24$