You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an alternative browser.
You should upgrade or use an alternative browser.
Algebra
- Thread starter smee
- Start date
D
Deleted member 4993
Guest
smee said:Home work for my kid - I put 4 pencils into each jar and have 1 jar left over. I put 3 pencils into each jar and have 1 pencil left over. How many jars and how many pencils ?
Need help urgently, Im quite retarded with things like this !!!
There are different ways do this problem - what grade level is your kid? Has s/he done Algebra - solution of linear equations?
\(\displaystyle \text{Hello, smee!}\)
\(\displaystyle \text{Here's one approach . . .}\)
\(\displaystyle \text{Let: }\:\begin{array}{ccc}J &=& \text{number of jars} \\ P &=& \text{number of penciles} \end{array}\)
\(\displaystyle \text{4 pencils per jar, 1 jar left over: }\quad\begin{array}{ccccccc} /\!\!/\!\!/\!\!/ & /\!\!/\!\!/\!\!/ & /\!\!/\!\1\!/\!\!/ && /\!\! /\!\! /\!\! / \\ \bigsqcup & \bigsqcup & \bigsqcup & \hdots & \bigsqcup && \bigsqcup\)
. . . . . . . . . . . . . . . . . . . . . . . . \(\displaystyle \underbrace{\qquad\qquad\qquad\qqiuad\qquad\quad\;}_{J-1\text{ jars}}\)
. . \(\displaystyle \text{We have: }\;P \;=\;4(J-1)\quad\bf{[1]}\)
\(\displaystyle \text{3 pencils per jar, 1 pencil left over: }\quad \begin{array}{ccccccc}/\!\! /\!\! / & /\!\! /\!\! / & /\!\! /\!\! / && / \!\! / \!\! / & / \\ \bigsqcup & \bigsqcup & \bigsqcup & \hdots & \bigsqcup\\ \end{array}\)
. . . . . . . . . . . . . . . . . . . . . . . . . .\(\displaystyle \underbrace{\qquad\qquad\qquad\qquad\quad\;}_{J\text{ jars}}\)
. . \(\displaystyle \text{We have: }\;P \:=\:3J + 1\quad\nf{[2]}\)
\(\displaystyle \text{Equate }\bf{[1]}\text{ and }\bf{[2]} \!:\) . \(\displaystyle 4(J-1) \:=\:3J + 1 \quad\Rightarrow\quad 4J - 4 \:=\:3J + 1 \quad\Rightarrow\quad J \:=\:5\)
\(\displaystyle \text{Substitute into }\nf{[2]}\!:\) . \(\displaystyle P \:=\:3(5)+1 \:=\:16\)
\(\displaystyle \text{Theefore, there are 5 jars and 16 pencils.}\)
\(\displaystyle \text{Here's one approach . . .}\)
I put 4 pencils into each jar and have 1 jar left over.
I put 3 pencils into each jar and have 1 pencil left over.
How many jars and how many pencils?
\(\displaystyle \text{Let: }\:\begin{array}{ccc}J &=& \text{number of jars} \\ P &=& \text{number of penciles} \end{array}\)
\(\displaystyle \text{4 pencils per jar, 1 jar left over: }\quad\begin{array}{ccccccc} /\!\!/\!\!/\!\!/ & /\!\!/\!\!/\!\!/ & /\!\!/\!\1\!/\!\!/ && /\!\! /\!\! /\!\! / \\ \bigsqcup & \bigsqcup & \bigsqcup & \hdots & \bigsqcup && \bigsqcup\)
. . . . . . . . . . . . . . . . . . . . . . . . \(\displaystyle \underbrace{\qquad\qquad\qquad\qqiuad\qquad\quad\;}_{J-1\text{ jars}}\)
. . \(\displaystyle \text{We have: }\;P \;=\;4(J-1)\quad\bf{[1]}\)
\(\displaystyle \text{3 pencils per jar, 1 pencil left over: }\quad \begin{array}{ccccccc}/\!\! /\!\! / & /\!\! /\!\! / & /\!\! /\!\! / && / \!\! / \!\! / & / \\ \bigsqcup & \bigsqcup & \bigsqcup & \hdots & \bigsqcup\\ \end{array}\)
. . . . . . . . . . . . . . . . . . . . . . . . . .\(\displaystyle \underbrace{\qquad\qquad\qquad\qquad\quad\;}_{J\text{ jars}}\)
. . \(\displaystyle \text{We have: }\;P \:=\:3J + 1\quad\nf{[2]}\)
\(\displaystyle \text{Equate }\bf{[1]}\text{ and }\bf{[2]} \!:\) . \(\displaystyle 4(J-1) \:=\:3J + 1 \quad\Rightarrow\quad 4J - 4 \:=\:3J + 1 \quad\Rightarrow\quad J \:=\:5\)
\(\displaystyle \text{Substitute into }\nf{[2]}\!:\) . \(\displaystyle P \:=\:3(5)+1 \:=\:16\)
\(\displaystyle \text{Theefore, there are 5 jars and 16 pencils.}\)
D
Deleted member 4993
Guest
Another way,
There are four pencils in each jar.
Then the number of pencils could be = 4, 8, 12, 16, 20, 24, 28.....
(There are atleast 2 jars - since 1 jar was left empty above)
There are three pencils in each Jar + 1. (There are atleast 2 jars - since 1 jar was left empty)
Then the number of pencils could be = 7, 10, 13, 16, 19, 22, 25.....
For a given number of jars - the only match is 5 jars and 16 pencils.
There are four pencils in each jar.
Then the number of pencils could be = 4, 8, 12, 16, 20, 24, 28.....
(There are atleast 2 jars - since 1 jar was left empty above)
There are three pencils in each Jar + 1. (There are atleast 2 jars - since 1 jar was left empty)
Then the number of pencils could be = 7, 10, 13, 16, 19, 22, 25.....
For a given number of jars - the only match is 5 jars and 16 pencils.
pencils per jar = a ; leftover jars = u
pencils per jar = b ; leftover pencils = v
JARS = (au + v) / (a - b)
I put 13 pencils into each jar and have 3 jars left over. [a = 13, u = 3]
I put 9 pencils into each jar and have 5 pencils left over. [b = 9, v = 5]
How many jars have I got?
Try that, Mommy Smee :wink:
pencils per jar = b ; leftover pencils = v
JARS = (au + v) / (a - b)
I put 13 pencils into each jar and have 3 jars left over. [a = 13, u = 3]
I put 9 pencils into each jar and have 5 pencils left over. [b = 9, v = 5]
How many jars have I got?
Try that, Mommy Smee :wink: