Proportions

Advanced Math 7 · Unit 14 · direct-inverse-proportion · Teacher edition

direct-inverse-proportion

Name ______________________________Date ______________

1.
$y$ is directly proportional to $x$ , and $y = 12$ when $x = 4$ . Find $y$ when $x = 7$ .
Answer: ______________
2.
The table shows pairs of values of $x$ and $y$ . Decide whether $y$ is directly or inversely proportional to $x$ , then find the missing value.

$x$ $2$ $4$ $10$
$y$ $30$ $15$ $?$

Enter the missing value of $y$ .
Answer: ______________
3.
A spring stretches $6$ cm when a $1.5$ kg mass hangs from it and $14$ cm when a $3.5$ kg mass hangs from it. The stretch $y$ (in cm) is directly proportional to the mass $x$ (in kg). Write the rule for $y$ in terms of $x$ .
Answer: ______________
4.
It takes $6$ painters $10$ days to paint a fence. Working at the same rate, how many days would it take $12$ painters?
1. (A)
  $2.5$ days
2. (B)
  $5$ days
3. (C)
  $20$ days
4. (D)
  $60$ days
5.
$a$ is directly proportional to $b$ , and $b$ is inversely proportional to $c$ . When $c = 4$ , $a = 6$ . What is $a$ when $c = 12$ ?
Answer: ______________

$x$	$2$	$4$	$10$
$y$	$30$	$15$	$?$

Answer key — Proportions

1.
21direct-inverse-proportion-01
$k = 12 \div 4 = 3$ , so $y = 3x$ and $y = 3 \times 7 = \mathbf{21}$ .
2.
6direct-inverse-proportion-02
$xy = 60$ in both complete columns, so the proportion is inverse and $y = \frac{60}{10} = \mathbf{6}$ . (Treating it as direct, $y = 15x$ , gives $150$ — but doubling $x$ from $2$ to $4$ halved $y$ .)
3.
$y=4x$ direct-inverse-proportion-03
$k = 6 \div 1.5 = 4$ (and $14 \div 3.5 = 4$ ✓), so $\mathbf{y = 4x}$ .
4.
(B)
$5$ days
direct-inverse-proportion-04
$6 \times 10 = 12 \times d \Rightarrow d = \mathbf{5}$ days. ( $20$ treats it as direct — “twice the painters, twice the days” — which is backwards; $60$ is the total painter-days, not the time.)
5.
2direct-inverse-proportion-05
$a = k_1 b = k_1 \cdot \frac{k_2}{c} = \frac{K}{c}$ , so $ac$ is constant: $ac = 24$ . At $c = 12$ , $a = \frac{24}{12} = \mathbf{2}$ . (Tripling $c$ divides $a$ by $3$ .)