W.W. Grainger (GWW) 10-Q 2012 Q1 Quarterly report

Exhibit 32

CERTIFICATION PURSUANT TO

18 U.S.C. SECTION 1350,

AS ADOPTED PURSUANT TO

SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

In connection with the Quarterly Report on Form 10-Q of W.W. Grainger, Inc. (“Grainger”) for the quarterly period ended March 31, 2012, (the “Report”), J. T. Ryan, as Chairman, President and Chief Executive Officer of Grainger, and R. L. Jadin, as Senior Vice President and Chief Financial Officer of Grainger, each hereby certifies, pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, that:

1. The Report fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934; and

2. The information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of Grainger.

/s/ J. T. Ryan

J. T. Ryan

Chairman, President and Chief Executive Officer

April 26, 2012

/s/ R. L. Jadin

R. L. Jadin

Senior Vice President and Chief Financial Officer

April 26, 2012